Quadratic Applications

Quadratic Projectile Problem	Linear Increase/Decrease Problem
Quadratic Trajectory (Path) Problem	Pythagorean Theorem Quadratic Application
Optimization of Area Problem	Quadratic Inequality Problem
Maximum Profit and Revenue Problems	Finding Quadratic Equation from Points or a Graph
Population Problem

Quadratic applications are very helpful in solving several types of word problems, especially where optimization is involved. Again, we can use the vertex to find the maximum or the minimum values, and roots to find solutions to quadratics.

Note that we did a Quadratic Inequality Application problem here in the Quadratic Inequalities section. Note also that we will discuss Optimization Problems using Calculus in the Optimization Using Calculus section.

Quadratic Projectile Problem:

Quadratic Projectile problems are common quadratic application problems. Here is an example:

Problem: Jennifer hit a golf ball from the ground and it followed the projectile $ h\left( t \right)=-16{{t}^{2}}+100t$, where $ t$ is the time in seconds, and $ h$ is the height of the ball. Find the highest point that her golf ball reached and also when it hits the ground again. Find a reasonable domain and range for this situation. Solution: Note that in this example, we are using the generic equation $ h\left( t \right)=-16{{t}^{2}}+{{v}_{0}}t+{{h}_{0}}$, where, in simplistic terms, –16 is the gravity (in feet per seconds per seconds), $ {{v}_{0}}$ is the initial velocity (in feet per seconds) and $ {{h}_{0}}$ is the initial height (in feet). (If units are in meters, the gravity is –4.9 meters per second per second). To get the highest point of the ball, use the vertex of the parabola. To get when the ball hits the ground again, set the height $ h$ to 0. Then use these two values to find a reasonable domain and range. Let’s try both algebraically and using a Graphing Calculator:
Solve Quadratic Algebraically	Solve Quadratic with Graphing Calculator
$ h\left( t \right)=-16{{t}^{2}}+100t$ To get the vertex, use ($ \displaystyle -\frac{b}{{2a}}$, plug $ \displaystyle -\frac{b}{{2a}}$ into the $ t$ to get the $ y$) to find the coordinates of the vertex, when $ y=a{{t}^{2}}+bt+c$: $ \displaystyle -\frac{b}{{2a}}=-\frac{{100}}{{-32}}=\,\,\,\,3.125\, \text{seconds}$ (This is the time, and to get the height, plug this into $ -16{{t}^{2}}+100t$, and get 156.25 feet.) To get when the ball hits the ground, set $ -16{{t}^{2}}+100t$ to 0; we get $ t=6.25$ seconds. This is the second root. This makes sense, since the ball started from the ground, so the parabola is symmetrical around the line of symmetry, which is $ x=3.125$.	Note that I used ZOOM 6, ZOOM 0, and ZOOM 3 ENTER a few times so I could see the vertex in the window. Then I used 2^nd TRACE (CALC), 4 (maximum), moved the cursor to the left of the top after “Left Bound?”, hit ENTER, moved the cursor to the right of the top after “Right Bound?”, and then hit ENTER twice to get the vertex. The $ y$-value of the vertex, the height, is 156.25 feet. To get the (positive) root, push 2^nd TRACE (CALC), and then push 2 for ZERO (or move cursor down to ZERO). The calculator will then say “Left Bound?” Using the cursors, move the cursor anywhere to the left of the zero (where the graph hits the $ x$-axis) and hit ENTER. When the calculator says “Right Bound?” move the cursor anywhere to the right of that zero and hit ENTER. The calculator will say “Guess?”. Hit ENTER once more, and you have the zero, which is 6.25 feet.
The reasonable domain is $ \left[ {0,6.25} \right]$ and the reasonable range is $ \left[ {0,156.25} \right]$, since time can’t be negative, and the ball can’t go beneath the ground.

(We will discuss projectile motion using parametric equations here in the Parametric Equations section.)

Quadratics Trajectory (Path) Problem

Note that typically, the independent variable represents time, not distance; however, sometimes parabolas represent the distance on the $ x$-axis and the height on the $ y$-axis, and the shapes are similar.

In these problems, the $ x$-axis is measuring the horizontal distance of the path of the ball, not the time, so the parabola is a true indication of the trajectory or path of the ball.

Quadratic Application Problem	Solution
Audrey throws a ball in the air, and the path the ball makes is modeled by the parabola $ y-8=-0.018{{\left( {x-20} \right)}^{2}}$, measured in feet. What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height? How far does the ball travel before it hits the ground?	Since the quadratic is basically already in vertex form ($ y=a{{\left( {x-h} \right)}^{2}}+k$, where $ (h,k)$ is the vertex), the vertex (from $ 0=-0.018{{\left( {x-20} \right)}^{2}}+8$) is $ (20,8)$. This means that the maximum height (since the parabola opens downward) is 8 feet and it happens 20 feet away from Audrey. When the ball hits the ground, $ y=0$, so $ 0=-0.018{{\left( {x-20} \right)}^{2}}+8$. We could expand the binomial and use the quadratic formula, but it’s much easier to use the square root method, since we have a square in the original: $ \begin{align}0&=-0.018{{\left( {x-20} \right)}^{2}}+8\\\frac{{-8}}{{-0.018}}&={{\left( {x-20} \right)}^{2}};\,\,\pm \sqrt{{\frac{8}{{0.018}}}}=\sqrt{{{{{\left( {x-20} \right)}}^{2}}}}\\x-20&\approx \pm 21.082\\x&\approx 41.082\end{align}$ Note that we had to “throw away” the negative part of the solution. The ball will hit the ground approximately 41.082 feet from Audrey. (We could have also used a graphing calculator to solve this problem.)
A kicker kicks a football in an attempt to make a field goal from 95 feet away from the uprights. The uprights are 14 feet high. The football reaches its maximum height of 75 feet when it is 50 feet from the kicker. What is the height of the ball when it reaches the uprights? Did the kicker make a field goal (did the ball go over the uprights?)	First sketch the problem; we want the height of the ball ($ y$) when it reaches the uprights : Get the equation of the quadratic for the path (height) of the ball, given the distance from the kicker. Since the vertex is at $ \left( {50,75} \right)$, $ y=a{{\left( {x-50} \right)}^{2}}+75$, and $ a$ will be negative, since the parabola faces down. Use the fact that $ \left( {0,0} \right)$ is a point to obtain $ a$: $ \begin{align}0&=a{{\left( {0-50} \right)}^{2}}+75\\-75&=2500a;\,\,\,\,a=-\frac{3}{{100}}\\y&=-\frac{3}{{100}}{{\left( {x-50} \right)}^{2}}+75\end{align}$ Now get the height of the ball when the distance from the kicker ($ x$) is 95 feet: $ \displaystyle y=-\frac{3}{{100}}{{\left( {95-50} \right)}^{2}}+75=14.25\,\text{feet}>14\,\text{feet}$. Yes! It barely goes over the upright, so the kicker made a goal!

Quadratic Application Problem

Solution

Audrey throws a ball in the air, and the path the ball makes is modeled by the parabola $ y-8=-0.018{{\left( {x-20} \right)}^{2}}$, measured in feet.

What is the maximum height the ball reaches, and how far (horizontally) from Audrey does is the ball at its maximum height?

How far does the ball travel before it hits the ground?

Since the quadratic is basically already in vertex form ($ y=a{{\left( {x-h} \right)}^{2}}+k$, where $ (h,k)$ is the vertex), the vertex (from $ 0=-0.018{{\left( {x-20} \right)}^{2}}+8$) is $ (20,8)$.

This means that the maximum height (since the parabola opens downward) is 8 feet and it happens 20 feet away from Audrey.

When the ball hits the ground, $ y=0$, so $ 0=-0.018{{\left( {x-20} \right)}^{2}}+8$. We could expand the binomial and use the quadratic formula, but it’s much easier to use the square root method, since we have a square in the original:

$ \begin{align}0&=-0.018{{\left( {x-20} \right)}^{2}}+8\\\frac{{-8}}{{-0.018}}&={{\left( {x-20} \right)}^{2}};\,\,\pm \sqrt{{\frac{8}{{0.018}}}}=\sqrt{{{{{\left( {x-20} \right)}}^{2}}}}\\x-20&\approx \pm 21.082\\x&\approx 41.082\end{align}$

Note that we had to “throw away” the negative part of the solution. The ball will hit the ground approximately 41.082 feet from Audrey. (We could have also used a graphing calculator to solve this problem.)

A kicker kicks a football in an attempt to make a field goal from 95 feet away from the uprights. The uprights are 14 feet high.

The football reaches its maximum height of 75 feet when it is 50 feet from the kicker.

What is the height of the ball when it reaches the uprights? Did the kicker make a field goal (did the ball go over the uprights?)

First sketch the problem; we want the height of the ball ($ y$) when it reaches the uprights :

Get the equation of the quadratic for the path (height) of the ball, given the distance from the kicker. Since the vertex is at $ \left( {50,75} \right)$, $ y=a{{\left( {x-50} \right)}^{2}}+75$, and $ a$ will be negative, since the parabola faces down. Use the fact that $ \left( {0,0} \right)$ is a point to obtain $ a$:

$ \begin{align}0&=a{{\left( {0-50} \right)}^{2}}+75\\-75&=2500a;\,\,\,\,a=-\frac{3}{{100}}\\y&=-\frac{3}{{100}}{{\left( {x-50} \right)}^{2}}+75\end{align}$

Now get the height of the ball when the distance from the kicker ($ x$) is 95 feet: $ \displaystyle y=-\frac{3}{{100}}{{\left( {95-50} \right)}^{2}}+75=14.25\,\text{feet}>14\,\text{feet}$. Yes! It barely goes over the upright, so the kicker made a goal!

Optimization of Area Problem:

A common application of quadratics is optimization, which typically involves finding the vertex of a parabola since it’s the highest or lowest amount. Here’s a problem:

Problem: A rectangular rose garden is being built against the back of a house with a fence around it, but there is only 120 feet of fencing available. What would be the dimensions (length and width) of the garden, with one side attached to the house, to make the area of the garden as large as possible? What is this maximum area? What is a reasonable domain for the width of the garden? Solution: Set the variables to what the problem is asking. As always, draw a picture first for any sort of “geometry” word problem. Since the sum of three sides of the garden is 120, if $ w$ equals the width, $ \displaystyle 2w+\text{length}=120$ (see picture). Thus, the length is $ 120-2w$, and the area is $ \text{width}\,\times \,\text{length}=w(120-2w)$. To maximize the area, use the vertex of this parabola: the vertex is the maximum $ \boldsymbol{y}$-point, given an $ \boldsymbol{x}$-point, (in our case, a $ \boldsymbol{w}$-point). Area depends on length and width – which makes sense. Get the vertex for $ A\left( w \right)=w\left( {120-2w} \right)$ both algebraically and using a Graphing Calculator:
Solve Quadratic Algebraically	Solve Quadratic with Graphing Calculator
$ A\left( w \right)=w\left( {120-2w} \right)$ To get the vertex, use ($ \displaystyle -\frac{b}{{2a}}$, plug $ \displaystyle -\frac{b}{{2a}}$ into the $ w$ to get the $ y$) to find the coordinates of the vertex, when $ y=a{{w}^{2}}+bw+c$: $ \displaystyle y=w\left( {120-2w} \right)\,\,\,\,\,\text{or}\,\,\,\,y=-2{{w}^{2}}+120w$ $ \displaystyle -\frac{b}{{2a}}=-\frac{{120}}{{-4}}=\,\,30\, \text{feet}$ This is the width; to get the length, $ 120-2w=120-2\left( {30} \right)=60\,\text{feet}$. Now plug in 30 ($ w$) to get the $ y$, or the area (could also have used simply $ 30\times 60$): $ \left( {30} \right)\left( {120-2\left( {30} \right)} \right)=30\left( {60} \right)=1800\, \text{fee}{{\text{t}}^{2}}$	Note that I used ZOOM 6, ZOOM 0, and ZOOM 3 ENTER a few times so I could see the vertex in the window. Then I used 2^nd TRACE (CALC), 4 (maximum), moved the cursor to the left of the top after “Left Bound?, moved the cursor to the right of the top after “Right Bound?”, and then hit ENTER twice to get the vertex. Since this is the $ w$ part of the vertex, 30 feet is the width that maximizes the area. The length is $ 120-2w$, or 60 feet. The area is length times width, or the $ y$-part of the vertex, which is $ 1800\text{ fee}{{\text{t}}^{2}}$.
To get the reasonable domain or appropriate domain, think about the values that the width could ever be – to make that garden at all. The width has to be positive, which means it has to be greater than zero. But there is a minus sign in one of the expressions containing the width (the length, which is $ 120-2w$), and this also must be positive. Since $ 120-2w>0,\,w<60$. A reasonable domain for the width is 0 to 60 feet or, in interval notation, $ \left( {0,60} \right)$. In this example, to find the reasonable range for the area, we could look at the graph, and see that it is $ \left( {0,1800} \right)$. Typically, a reasonable range for these types of problems is 0 to the $ \boldsymbol{y}$-portion of the vertex.

Maximum Profit Problem:

Here’s another optimization problem:

Quadratic Maximum Profit Problem	Solution
The profit from selling local ballet tickets depends on the ticket price. Using past receipts, the profit can be modeled by the function $ p=-15{{x}^{2}}+600x+60$, where $ x$ is the price of each ticket. What is the ticket price that gives the maximum profit, and what is that maximum profit?	This problem is more straightforward, since we are given a profit formula. Either graph the function to get the vertex, or use ($ \displaystyle -\frac{b}{{2a}}$, plug $ \displaystyle -\frac{b}{{2a}}$ into the $ x$ to get the $ y$) to find the coordinates of the vertex: $ \displaystyle -\frac{b}{{2a}}=-\frac{{600}}{{2\left( {-15} \right)}}=20$ $ \displaystyle f\left( {20} \right)=-15{{\left( {20} \right)}^{2}}+600\left( {20} \right)+60=6060$ Since the vertex is $ \left( {20,6060} \right)$, the ticket price should be $20 to maximize profit, and that maximum profit is $6060.

Quadratic Maximum Profit Problem

Solution

The profit from selling local ballet tickets depends on the ticket price. Using past receipts, the profit can be modeled by the function $ p=-15{{x}^{2}}+600x+60$, where $ x$ is the price of each ticket.

What is the ticket price that gives the maximum profit, and what is that maximum profit?

This problem is more straightforward, since we are given a profit formula.

Either graph the function to get the vertex, or use ($ \displaystyle -\frac{b}{{2a}}$, plug $ \displaystyle -\frac{b}{{2a}}$ into the $ x$ to get the $ y$) to find the coordinates of the vertex:

$ \displaystyle -\frac{b}{{2a}}=-\frac{{600}}{{2\left( {-15} \right)}}=20$

$ \displaystyle f\left( {20} \right)=-15{{\left( {20} \right)}^{2}}+600\left( {20} \right)+60=6060$

Since the vertex is $ \left( {20,6060} \right)$, the ticket price should be $20 to maximize profit, and that maximum profit is $6060.

Maximum Revenue Problem:

Here’s another optimization problem:

Quadratic Maximum Revenue Problem	Solution
A popular designer purse sells for $500 and 45,000 are sold a month. The company did some research and realized that for each $20 decrease in price, they can sell 5000 more purses per month. How much should the company charge for the purse so they can maximize monthly revenues?	This problem is a little trickier since we can’t really tell from the question what the variables should be. Start with what we know. Think of monthly revenue (what a company makes each month) as “price times number of purses sold”. For example, without making any changes, the monthly revenue is $ \$500\times 45,000=\$22,500,000$. But, with the new information, we know that for each $20 decrease in price, they can sell an increase of 5000 purses per month. For example, if they sell the purses for $ \$500-\$20=\$480$, they would sell $ 45,000+5,000=50,000$ purses, for a monthly revenue of $24,000,000, which is more! Use algebra to find the maximum monthly revenue by letting $ x=$ the number of $20 decreases (and hence sales of 5000 more purses) per month. Then find the maximum point (vertex) of the quadratic to get the maximum monthly revenue. Here is the equation: $ \displaystyle \begin{align}\text{revenue}&=\,\,\,\,\,\left( {\text{price}} \right)\left( {\text{number sold}} \right)\\y&=\left( {500-20x} \right)\left( {45000+5000x} \right)\end{align}$ Find the vertex (maximum) of this quadratic equation, which we can get from a graphing calculator $ \left( {8,28900000} \right)$. Thus, 8 is the number of $20 decreases the company can charge a month (which is the number of “batches of 5000 more purses” the company can sell). This means to get the maximum revenues, the company should sell their purses for $ \$500-\$20(8)=\$500-\$160=\$340$ each, to sell $ 45,000+5000\left( 8 \right)=85,000$ purses per month, for a maximum profit of $28,900,000.

Quadratic Maximum Revenue Problem

Solution

A popular designer purse sells for $500 and 45,000 are sold a month. The company did some research and realized that for each $20 decrease in price, they can sell 5000 more purses per month.

How much should the company charge for the purse so they can maximize monthly revenues?

This problem is a little trickier since we can’t really tell from the question what the variables should be. Start with what we know. Think of monthly revenue (what a company makes each month) as “price times number of purses sold”. For example, without making any changes, the monthly revenue is $ \$500\times 45,000=\$22,500,000$.

But, with the new information, we know that for each $20 decrease in price, they can sell an increase of 5000 purses per month. For example, if they sell the purses for $ \$500-\$20=\$480$, they would sell $ 45,000+5,000=50,000$ purses, for a monthly revenue of $24,000,000, which is more!

Use algebra to find the maximum monthly revenue by letting $ x=$ the number of $20 decreases (and hence sales of 5000 more purses) per month. Then find the maximum point (vertex) of the quadratic to get the maximum monthly revenue. Here is the equation:

$ \displaystyle \begin{align}\text{revenue}&=\,\,\,\,\,\left( {\text{price}} \right)\left( {\text{number sold}} \right)\\y&=\left( {500-20x} \right)\left( {45000+5000x} \right)\end{align}$

Find the vertex (maximum) of this quadratic equation, which we can get from a graphing calculator $ \left( {8,28900000} \right)$. Thus, 8 is the number of $20 decreases the company can charge a month (which is the number of “batches of 5000 more purses” the company can sell). This means to get the maximum revenues, the company should sell their purses for $ \$500-\$20(8)=\$500-\$160=\$340$ each, to sell $ 45,000+5000\left( 8 \right)=85,000$ purses per month, for a maximum profit of $28,900,000.

Bunny Rabbit Population Problem:

Quadratic Application Problem	Solution and Calculator Instructions
The observed bunny rabbit population on an island is given by the function $ p=-.4{{t}^{2}}+130t+1200$, where $ t$ is the time in months since they began observing the rabbits. (a) When is the maximum population attained? (b) What is the maximum population? (c) When does the bunny rabbit population disappear from the island?	Find the vertex on the graphing calculator the same way we did above using the 2nd trace (Calc) Maximum feature: the vertex is $ \left( {162.5,11762.5} \right)$. The maximum rabbit population was roughly 11762 rabbits (we can’t have half of a rabbit!) when it was 162.5 months after they began observing the rabbit population. This answers (a) and (b). For (c), we need to see when the graph goes back down to 0; this is when there are no rabbits left on the island. To get the roots, push 2^nd TRACE (CALC), and then push 2 for ZERO (or move cursor down to ZERO). The calculator will then say “Left Bound?” Using the cursors, move the cursor anywhere to the left of the zero (where the graph hits the $ x$-axis) and hit ENTER. We want the zero that is positive. When the calculator says “Right Bound?” move the cursor anywhere to the right of that zero and hit ENTER. The calculator will say “Guess?”. Hit ENTER once more, and you have your zero. Note that sometimes the calculator gets confused and gives you a number way close to 0 (-1E-9) instead of 0 for the $ y$-value. So, for answer (c) , the rabbit population will disappear from the island at around 334 months from when the observations started. Note that you can also find the roots by setting “$ {{Y}_{2}}=$ ” to 0 and use the Intercept function to find the roots. You may have to use the ZOOM and/or WINDOWS to make sure you see the point of intersection you want. You may also have to use TRACE and then arrows to move the cursor close to the point of intersection you want (the positive one), if the cursor is closer to the other root. To get the point(s) of intersection, push 2^nd TRACE (CALC), and then either push 5, or move cursor down to intersect. You should see “First curve?” at the bottom. Then push ENTER, ENTER, ENTER:

Quadratic Application Problem

Solution and Calculator Instructions

The observed bunny rabbit population on an island is given by the function $ p=-.4{{t}^{2}}+130t+1200$, where $ t$ is the time in months since they began observing the rabbits.

(a) When is the maximum population attained?

(b) What is the maximum population?

Find the vertex on the graphing calculator the same way we did above using the 2nd trace (Calc) Maximum feature: the vertex is $ \left( {162.5,11762.5} \right)$. The maximum rabbit population was roughly 11762 rabbits (we can’t have half of a rabbit!) when it was 162.5 months after they began observing the rabbit population. This answers (a) and (b).

For (c), we need to see when the graph goes back down to 0; this is when there are no rabbits left on the island. To get the roots, push 2^nd TRACE (CALC), and then push 2 for ZERO (or move cursor down to ZERO). The calculator will then say “Left Bound?” Using the cursors, move the cursor anywhere to the left of the zero (where the graph hits the $ x$-axis) and hit ENTER. We want the zero that is positive. When the calculator says “Right Bound?” move the cursor anywhere to the right of that zero and hit ENTER. The calculator will say “Guess?”. Hit ENTER once more, and you have your zero. Note that sometimes the calculator gets confused and gives you a number way close to 0 (-1E-9) instead of 0 for the $ y$-value.

So, for answer (c) , the rabbit population will disappear from the island at around 334 months from when the observations started.

Note that you can also find the roots by setting “$ {{Y}_{2}}=$ ” to 0 and use the Intercept function to find the roots. You may have to use the ZOOM and/or WINDOWS to make sure you see the point of intersection you want. You may also have to use TRACE and then arrows to move the cursor close to the point of intersection you want (the positive one), if the cursor is closer to the other root. To get the point(s) of intersection, push 2^nd TRACE (CALC), and then either push 5, or move cursor down to intersect. You should see “First curve?” at the bottom. Then push ENTER, ENTER, ENTER:

Linear Increase/Decrease Problem:

OK, use your imaginations on this one 🙂

Problem:

Taylor and Miranda are performing on a magic dimension-changing stage that is 20 yards long by 15 yards wide. The length is decreasing linearly (with time) at a rate of 2 yards per hour, and the width is increasing linearly (with time) at a rate of 3 yards per hour. When will the stage have the maximum area, and when will the stage disappear (has an area of 0 square yards)?

Solution:

This one’s a little trickier, since we are asking when the stage will be the greatest area, and when it will have an area of 0, yet we are only given distances and rates. Since we’re finding areas, we need to work with distances only. We know that $ \text{Distance}=\text{Rate}\times \text{Time}$. Do you see how at time $ t$, the length of the stage is $ (20-2t)$ and the width is $ (15+3t)$? Think about it: after one hour, the length of the stage will have decreased by 2 yards, and the width will have increased by 3 yards, so the new stage will be 18 by 18 yards. After two hours, the length will be 16 yards, and the width will be 21 yards, and so on. Answer the questions, with and without the graphing calculator:

Solve Quadratic Algebraically

Solve Quadratic with Graphing Calculator

$ A\left( t \right)=\left( {20-2t} \right)\left( {15+3t} \right)$

To get the vertex, we can use $ \displaystyle -\frac{b}{{2a}}$ ($ \displaystyle -\frac{b}{{2a}}$, plug into the $ t$ to get the $ A(t)$ or $ y$) to find the coordinates of the vertex, when $ y=a{{t}^{2}}+bt+c$:

$ y=\left( {20-2t} \right)\left( {15+3t} \right)\,\,\,\,\text{or}\,\,\,y=-6{{t}^{2}}+30t+300$

$ \displaystyle \,\,\frac{{-b}}{{2a}}=\frac{{-30}}{{-12}}=\,2.5 \text{ hours}$

This is the time that the area of the stage will be at a maximum.

Now plug in 2.5 ($ t$) to get the $ y$, or the area:

$ \begin{align}A\left( t \right)&=-6{{t}^{2}}+30t+300\\&=-6{{\left( {2.5} \right)}^{2}}+30\left( {2.5} \right)+300\\&=337.5 \text{ yard}{{\text{s}}^{\text{2}}}\end{align}$

Note that I used ZOOM 6, ZOOM 0, and ZOOM 3 ENTER a few times so I could see the vertex in the window.

Use 2^nd TRACE (CALC), 4 (maximum), move the cursor to the left of the top after “Left Bound?”, move the cursor to the right of the top after “Right Bound?”, and then hit ENTER twice to get the vertex.

Since this is the “$ t$” part of the vertex, 2.5 hours is the time that maximizes the area.

The area is the “$ y$” part of the vertex, which is $ 337.5$ yards².

Now find when the stage will have no area left. Set the equation to 0, or find the rightmost root with the calculator:

$ 0=\left( {20-2t} \right)\left( {15+3t} \right)$

Since this is already in factored form, we can just set each factor to 0 to see when the quadratic will be 0:

$ \displaystyle \begin{array}{l}20-2t=0\,\,\,\,\,\,15+3t=0\\\,\,\,\,20=2t\,\,\,\,\,\,\,\,\,15=-3t\,\,\,\\\,\,\,\,\,t=10\,\,\,\,\,\,\,\,\,\,\,\,t=-5\end{array}$

Since we can’t have a negative time, “throw away” the –5 root.

The time that the area will be 0 is in 10 hours.

To get the roots, push “2^nd TRACE” (CALC), and then push 2 for ZERO (or move cursor down to ZERO).

The calculator will then say “Left Bound?” Using the cursors, move the cursor anywhere to the left of the zero (where the graph hits the $ x$-axis) and hit ENTER. We want the zero that is positive (rightmost zero). When the calculator says “Right Bound?” move the cursor anywhere to the right of that zero and hit ENTER.

The calculator will say “Guess?”. Hit ENTER once more, and you have your zero. We see that $ x=10$; the time the area will be zero is in 10 hours.

Pythagorean Theorem Quadratic Application:

Here’s one that uses the Pythagorean Theorem:

Quadratic Problem and Picture	Solution
The hypotenuse of a right triangle is 4 inches longer than one leg and 2 inches longer than the other. Find the dimensions of the triangle. Also, find a reasonable domain for the hypotenuse. I decided to make the hypotenuse of the triangle the $ x$-value since it was easier to get a reasonable domain for the hypotenuse:	Use the Pythagorean Theorem to set up the problem: the sum of the squares of the two legs (the sides next to the right angle) adds up to the square of the hypotenuse: $ {{\left( {x-2} \right)}^{2}}+{{\left( {x-4} \right)}^{2}}={{x}^{2}}$. Multiply out (FOIL) the binomials and put everything to one side and factor to get the answers: $ \begin{array}{c}\,{{\left( {x-2} \right)}^{2}}+{{\left( {x-4} \right)}^{2}}={{x}^{2}}\\{{x}^{2}}-4x+4+{{x}^{2}}-8x+16={{x}^{2}}\\{{x}^{2}}-12x+20=0\\\left( {x-10} \right)\left( {x-2} \right)=0\\x=2,\,\,10\end{array}$ Now, look at the reasonable domain for the hypotenuse. All sides of the triangle have to be greater than 0, so (positive), so look at the minus signs in the expressions for the legs: “$ x-2$”, and “$ x-4$”. Both must be positive, so $ x$ has to be greater than 4 (anything less would result in a negative leg value). Therefore, the reasonable domain for the hypotenuse is $ x>4$, or $ \left( {4,\infty } \right)$. Only the 10-value is in the domain of $ \left( {4,\infty } \right)$; thus, $ x$-value is 10 inches (hypotenuse), and the sides are 8 inches and 6 inches. Check our answers: $ {{8}^{2}}+{{6}^{2}}={{10}^{2}}.\,\,\,\,\surd $

Quadratic Problem and Picture

Solution

The hypotenuse of a right triangle is 4 inches longer than one leg and 2 inches longer than the other. Find the dimensions of the triangle.

Also, find a reasonable domain for the hypotenuse.

I decided to make the hypotenuse of the triangle the $ x$-value since it was easier to get a reasonable domain for the hypotenuse:

Use the Pythagorean Theorem to set up the problem: the sum of the squares of the two legs (the sides next to the right angle) adds up to the square of the hypotenuse: $ {{\left( {x-2} \right)}^{2}}+{{\left( {x-4} \right)}^{2}}={{x}^{2}}$.

Multiply out (FOIL) the binomials and put everything to one side and factor to get the answers:

$ \begin{array}{c}\,{{\left( {x-2} \right)}^{2}}+{{\left( {x-4} \right)}^{2}}={{x}^{2}}\\{{x}^{2}}-4x+4+{{x}^{2}}-8x+16={{x}^{2}}\\{{x}^{2}}-12x+20=0\\\left( {x-10} \right)\left( {x-2} \right)=0\\x=2,\,\,10\end{array}$

Now, look at the reasonable domain for the hypotenuse. All sides of the triangle have to be greater than 0, so (positive), so look at the minus signs in the expressions for the legs: “$ x-2$”, and “$ x-4$”. Both must be positive, so $ x$ has to be greater than 4 (anything less would result in a negative leg value). Therefore, the reasonable domain for the hypotenuse is $ x>4$, or $ \left( {4,\infty } \right)$.

Only the 10-value is in the domain of $ \left( {4,\infty } \right)$; thus, $ x$-value is 10 inches (hypotenuse), and the sides are 8 inches and 6 inches. Check our answers: $ {{8}^{2}}+{{6}^{2}}={{10}^{2}}.\,\,\,\,\surd $

Quadratic Inequality Problem:

You may encounter a Quadratic Inequality problem like this:

Quadratic Discriminant Problem	Solution
Given $ 2{{x}^{2}}+bx+8$, find the values of $ b$ so there are no real solutions to the quadratic.	We learned here in the Introduction to Quadratics section that the value of the discriminant of quadratics in standard form $ a{{x}^{2}}+bx+c$ has three possibilities: $ \displaystyle {{b}^{2}}-4ac=0$ means there is only one real solution. $ \displaystyle {{b}^{2}}-4ac>0$ means there are two real solutions. $ \displaystyle {{b}^{2}}-4ac<0$ means there are no real solutions, only imaginary solutions. In the quadratic given, $ a=2$ and $ c=8$. Find $ b$ such that the discriminant is $ <0$: $ \displaystyle \begin{array}{c}{{b}^{2}}-4\left( 2 \right)\left( 8 \right)<0\\{{b}^{2}}-64<0\\{{b}^{2}}<64\\\left\| b \right\|<8\end{array}$ $ \displaystyle \begin{array}{c}b<8\,\,\,\,\,\,\,\text{and}\,\,\,\,\,b>-8\\b>-8\,\,\,\,\,\,\text{and}\,\,\,\,\,b<8\\-8<x<8\\\left( {-8,8} \right)\end{array}$ $ b$ would have to be between –8 and 8 (but can’t include –8 or 8) so there are no real solutions to $ 2{{x}^{2}}+bx+8$. Try some numbers for $ b$ to convince yourself that this is correct!

Quadratic Discriminant Problem

Solution

Given $ 2{{x}^{2}}+bx+8$, find the values of $ b$ so there are no real solutions to the quadratic.

We learned here in the Introduction to Quadratics section that the value of the discriminant of quadratics in standard form $ a{{x}^{2}}+bx+c$ has three possibilities:

$ \displaystyle {{b}^{2}}-4ac=0$ means there is only one real solution.
$ \displaystyle {{b}^{2}}-4ac>0$ means there are two real solutions.
$ \displaystyle {{b}^{2}}-4ac<0$ means there are no real solutions, only imaginary solutions.

In the quadratic given, $ a=2$ and $ c=8$. Find $ b$ such that the discriminant is $ <0$:

$ \displaystyle \begin{array}{c}{{b}^{2}}-4\left( 2 \right)\left( 8 \right)<0\\{{b}^{2}}-64<0\\{{b}^{2}}<64\\\left| b \right|<8\end{array}$

$ \displaystyle \begin{array}{c}b<8\,\,\,\,\,\,\,\text{and}\,\,\,\,\,b>-8\\b>-8\,\,\,\,\,\,\text{and}\,\,\,\,\,b<8\\-8<x<8\\\left( {-8,8} \right)\end{array}$

$ b$ would have to be between –8 and 8 (but can’t include –8 or 8) so there are no real solutions to $ 2{{x}^{2}}+bx+8$. Try some numbers for $ b$ to convince yourself that this is correct!

Finding Quadratic Equations from Points or a Graph

We saw how to obtain a Quadratic Equation from a point and/or graph here in the Solving Quadratics by Factoring and Completing the Square section. Here is a problem addressing this:

Solution:

Quadratic Problem and Graph	Solution
Emmy throws a dog toy up in the air from 5 feet above the ground. When the toy is 2 feet from the her, the toy reaches a maximum height of 9 feet, and then lands back on the ground 5 feet from her. Find the “$ a$” (the coefficient of the $ {{x}^{2}}$) for the parabola of the flight of the toy, and write this quadratic equation in vertex form, standard form, and factored form.	Draw a graph; notice that the vertex is the highest point, the $ y$-intercept is where she starts throwing the toy, and the positive $ x$-intercept is where the toy lands. Since the vertex (highest point) is $ (2,9)$, the vertex form of the equation $ y=a{{\left( {x-2} \right)}^{2}}+9$. “$ a$” will be negative, since the parabola faces downwards. Use the $ y$-intercept $ (0,5)$ to plug in to $ (x,y)$ to get “$ a$”: $ \begin{align}y&=a{{\left( {x-2} \right)}^{2}}+9\\5&=a{{\left( {0-2} \right)}^{2}}+9;\,\,5=4a+9\\-4&=4a;\,\,\,\,\,a=-1\end{align}$ The vertex form is $ y=-1{{\left( {x-2} \right)}^{2}}+9$ (or $ y=-{{\left( {x-2} \right)}^{2}}+9$), and the standard form (by multiplying it out) is $ y=-{{x}^{2}}+4x+5$. Factor this to get the factored form, which is $ y=-\left( {x-5} \right)\left( {x+1} \right)$ (take the negative out first). You can check all three forms by putting them all in a graphing calculator (like in $ \displaystyle {{Y}_{1}},\,{{Y}_{2}},\,{{Y}_{3}}$) to make sure they are all the same parabola! (You can see from the graph that this looks correct; the roots look like $ (0,-1)$ and $ (5,0)$.)

Quadratic Problem and Graph

Solution

Emmy throws a dog toy up in the air from 5 feet above the ground. When the toy is 2 feet from the her, the toy reaches a maximum height of 9 feet, and then lands back on the ground 5 feet from her.

Find the “$ a$” (the coefficient of the $ {{x}^{2}}$) for the parabola of the flight of the toy, and write this quadratic equation in vertex form, standard form, and factored form.

Draw a graph; notice that the vertex is the highest point, the $ y$-intercept is where she starts throwing the toy, and the positive $ x$-intercept is where the toy lands.

Since the vertex (highest point) is $ (2,9)$, the vertex form of the equation $ y=a{{\left( {x-2} \right)}^{2}}+9$. “$ a$” will be negative, since the parabola faces downwards.

Use the $ y$-intercept $ (0,5)$ to plug in to $ (x,y)$ to get “$ a$”:

$ \begin{align}y&=a{{\left( {x-2} \right)}^{2}}+9\\5&=a{{\left( {0-2} \right)}^{2}}+9;\,\,5=4a+9\\-4&=4a;\,\,\,\,\,a=-1\end{align}$

The vertex form is $ y=-1{{\left( {x-2} \right)}^{2}}+9$ (or $ y=-{{\left( {x-2} \right)}^{2}}+9$), and the standard form (by multiplying it out) is $ y=-{{x}^{2}}+4x+5$.

Factor this to get the factored form, which is $ y=-\left( {x-5} \right)\left( {x+1} \right)$ (take the negative out first).

You can check all three forms by putting them all in a graphing calculator (like in $ \displaystyle {{Y}_{1}},\,{{Y}_{2}},\,{{Y}_{3}}$) to make sure they are all the same parabola! (You can see from the graph that this looks correct; the roots look like $ (0,-1)$ and $ (5,0)$.)

Learn these rules, practice, practice, practice, and you’ll rock at math!

On to Imaginary (Non-Real) and Complex Numbers – you’re ready!