Now that we’ve learned about **Quadratics** and **Factoring** and did some work with square roots, we can go back and revisit **solving radical equations and inequalities**, with special emphasis on **square root functions**. We solved some radical equations in the **Solving Exponential and Radical Equations** portion in the **Exponents and Radicals in Algebra** section, but now we can work with more complicated equations where we can multiply binomials to find the answers!

For working with **inverse functions of radicals**, see the **Inverses of Functions** section. For **factoring and solving with Exponents**, see the **Exponential Functions** section.

## Parent Graphs of Radical Functions

First of all, let’s see what some basic radical function graphs look like. The first set of graphs are the quadratics and the square root functions; since the square root function “undoes” the quadratic function, it makes sense that it looks like a quadratic on its side. But the important thing to note about the simplest form of the square root function $ y=\sqrt{x}$ is that the **range** ($ y$) **by definition is only positive**; thus we only see “half” of a sideways parabola. The **domain **($ x$)** is **always positive, too, since we can’t take the square root of a negative number.

Remember also that another way to write $ y=\sqrt{x}$ is $ y={{x}^{{\frac{1}{2}}}}$.

See how, since $ {{2}^{2}}=4$, a point on the quadratic graph is $ \left( {2,4} \right)$? Similarly, since $ \sqrt{4}=2$, a point on the square root graph is $ \left( {4,2} \right)$.

Next, we have the **cubic** (raising something to the 3^{rd} power) and **cube root function** graphs. Since cube roots can be **both positive and negative**, the domain and range of both graphs is the **set of real numbers**.

Remember also that another way to write $ y=\sqrt[3]{x}$ is $ y={{x}^{{\frac{1}{3}}}}$.

See how, since $ {{2}^{3}}=8$, a point on the cube function graph is $ \left( {2,8} \right)$ (and so is $ \left( {-2,-8} \right)$)? Similarly, since $ \sqrt[3]{8}=2$, a point on the cube root graph is $ \left( {8,2} \right)$. Note that $ \left( {-8,-2} \right)$ is also a point on this graph, since $ \sqrt[3]{-8}=-2$.

We’ll talk a little later in the **Inverses of Functions** section that the quadratic and square root functions are “opposites” or **inverses** of each other. The cubic and cube root functions are also **inverses** of each other.

## Transformations of Radical Functions

We will learn more about **Transformations of Parent Functions** in the **Parent Functions and Transformations** section.

The generic equation for a transformation with vertical stretch $ a$, horizontal stretch $ b$, horizontal shift $ h$, and vertical shift $ k$ is $ \displaystyle f\left( x \right)=a\,\sqrt[n]{{\frac{1}{b}\left( {x-h} \right)}}+k$ for radical functions.

**Remember these rules**:

When functions are transformed on the **outside **of the $ f(x)$ part, you move the function **up and down** and do “**regular**” math, as we’ll see in the examples below. These are **vertical** **transformations **o**r translations**.

When transformations are made on the **inside** of the $ f(x)$ part, you move the function **back and forth** (but do the **opposite** **math** – basically since if you were to isolate the $ x$, you’d move everything to the other side). These are **horizontal** **transformations** or **translations**.

When there is a **negative sign** outside the parentheses, the function is reflected (flipped) across the $ x$-axis; when there is a negative sign inside the parentheses, the function is reflected across the $ y$-axis.

Here are some examples, using t-charts:

## Solving Radical Equations Algebraically

Now let’s solve some problems with square root functions. With even radicals, we have to make sure that **our answers never produce a negative number underneath the square root** (even radical) **sign**. Also, **if we raise both sides to an even exponent (like squaring), we need to check our answers**, since some solutions may not work. Both of these conditions can produce **extraneous solutions** (solutions that don’t work), since even exponents can be a little quirky.

The main idea in solving these is to **get rid of the radical signs by raising both sides to that exponent**. For example, with square roots, we have to square both sides. If we have two square roots, it’s easiest to have them separated so when we square both sides, it’s not as complicated. Sometimes we have to take the square of each side more than once, after we’ve FOILED one or both sides (see last example in next set of examples).

Also remember that you can always turn a radical into a rational (fractional) exponent; here is an example: $ {{\left( {\sqrt[3]{x}} \right)}^{4}}=\sqrt[3]{{{{x}^{4}}}}={{x}^{{\frac{4}{3}}}}$.

And don’t forget when we end up with a **quadratic equation**, put everything on one side (**set to 0**) and factor, or use the quadratic formula.

Here are some of the problems we solved previously and a few more (since we know how to FOIL now!):

Here are a few where we have to square **both sides** **two times **to get rid of the radicals. Note the second problem has a **radical inside of a radical**.

And are a couple of examples with **odd-indexed radicals**, where we can sit back and relax and just solve – everything we get should work!

## Solving Radical Equations Graphically

We can graph radical functions either with a *t*-chart or in the **graphing calculator**. Later, we’ll learn how to transform functions more easily in the **Parent Graphs and Transformations** section.

Since we’re so good with the graphing calculator (yeah!), let’s solve a radical function equation using the calculator:

## Solving Radical Inequalities Algebraically

Note that we saw some of these same examples in the **Exponents and Radicals in Algebra** section.

**Remember these rules for solving inequalities algebraically**:

- When solving inequalities, we need to be careful with
**multiplying and dividing by anything negative**, where we have to**change the direction of the inequality sign**. - What’s under an
**even radical**has to be**positive**(**domain restriction**); we have to**create another inequality**and set what’s**under the even radical to**$ \boldsymbol{{\ge 0}}$. We then solve for $ x$, and take the**intersection**of both solutions. The reason we take the intersection of the two solutions is because**both**must work. - For more advanced solving, we’ll want to use a
**sign chart**to show the**intervals that work and don’t work**; when we solve for $ x$ in these situations, we get the**critical values**for the sign chart. We then have to check each interval to see if the inequality is true or false. - For
**radicals**, when there is a variable not under a square root and we square both sides, we have to be careful since don’t know if the side without the square is positive or negative (and thus if we should switch the sign). In these cases, check the interval test values in the original inequality, and use**T**or**F**(or**Y**or**N**) to indicate whether or not they work. - If we get something like $ \sqrt{n}<0$ (or a negative number), there is
**no solution**, and something like $ \sqrt{n}\ge 0$ (or a negative number), we get all real numbers, except for the domain restriction ($ n\ge 0$). - You can check these inequalities in your graphing calculator to make sure they are correct. For example, for the first one use $ {{Y}_{1}}=\sqrt{{{{x}^{2}}-2x-8}}$ and $ {{Y}_{2}}=x+2$.

Here are some examples; note that we just **raise each side to the root to get rid of it**. We can do that in these examples, **since we know the sign of the values on both sides. **It gets trickier when we don’t know the sign of one of the sides.

Here are more complicated problems where we need to use a **sign chart** to solve radical inequalities.

**Note that we have to be careful when there is a variable on a side and it’s not under a square root; when we square both sides, we’re not really sure if we’d have to switch the inequality sign. This is because we don’t know if the side without the square is positive or negative.** Since this is the case, it’s best to check the intervals in the **original inequality**, and use **T** or **F** (or **Y** or **N**) when checking.

Here are a few more problems with radical inequalities:

## Solving Radical Inequalities Graphically

### Graphing Radical Inequalities

Let’s first just graph a simple **radical inequality** to show what the shading looks like; you may have to make some graphs like this. We saw earlier what the radical function looks like, and we can use the “rain up” (for $ >$) and “rain down” (for $ <$) shading like we did here in the** Quadratics Inequalities** section. Remember that with “$ <$” and “$ >$” inequalities, we draw a **dashed (or dotted) line** to indicate that we’re not really including that line (but everything up to it), whereas with “$ \le $” and “$ \ge $”, we draw a **regular line**, to indicate that we are including it in the solution.

Note that we also had to check so that **anything under the even radical** **is** **positive**; this is why the graph is shaded for $ x\ge 5$. We still have to keep this vertical line dotted, since we take the intersection (both have to work) of the two inequalities, and in this example, we have $ <$.

Note that we can put this in the **graphing calculator**, too. We had to move the cursor way to the left of “$ {{\text{Y}}_{1}}=$” and change to an inequality (play around with it; it’s different with the color calculator!) and then graph:

### Solving Radical Inequalities Graphically

We can also **solve radical inequalities graphically**. To get the intervals for $ x$ for these graphs, you have to look and see whichever graph is **on the bottom or below the other one **(has the smaller $ y$ for that interval) if it is a “**less than**” problem. For a “**greater than**” problem, you find the interval of the graph that is **on the top or above the other one **(has a larger $ y$ for that interval).

Sometimes (like in the third example below), there are no values that make the inequality true.

**NOTE**: We could solve use **graphing calculator** as we did for the equalities above, but it’s really difficult to get the point of intersection where the graphs hit the $ x$-**axis**, since the graphs are just starting there (you might use the table). Also, when you get the other points of intersection, it’s easiest if you have use **TRACE** to move the cursor **above the intersection of the functions**, if there is an intersection. When you get the point of intersection, use the $ x$value, since we’re solving for $ x$.

**Learn these rules, and practice, practice, practice!**

**For Practice**: Use the **Mathway** widget below to try an **Inequality** problem. Click on **Submit** (the blue arrow to the right of the problem) and click on **Solve for x** to see the answer.

You can also type in your own problem, or click on the three dots in the upper right hand corner and click on “Examples” to drill down by topic.

If you click on **Tap to view steps**, or **Click Here**, you can register at **Mathway** for a **free trial**, and then upgrade to a paid subscription at any time (to get any type of math problem solved!).

On to **Solving Inequalities** – you are ready!