# Part 1

In this video, we are going to solve equations with a radical expression.

• Only $19.95/mo. No commitment. Cancel anytime. In $\sqrt{x}=7$, first square both sides of the equation $\sqrt{x}^2=7^2$ $x=49$ If there are other values in the equation, then first isolate the $\sqrt{x}$ before squaring both sides of the equation. ## Video-Lesson Transcript Let’s go over how to solve equations with a radical expression. We have $\sqrt{x} = 7$ In order to solve for $x$, we have to get rid of the radical sign. The inverse of a square root is squaring. So we’ll square both sides $(\sqrt{x})^2 = 7^2$ $x = 49$ If we have something like this: $\sqrt{x} + 3 = 8$ we want to get the radical by itself first. Let’s subtract $3$ on both sides $\sqrt{x} + 3 - 3 = 8 - 3$ $\sqrt{x} = 5$ And the last step is to square both sides. $(\sqrt{x})^2 = 5^2$ $x = 25$ Let’s look at a more complicated one $3 \sqrt{x} - 4 = 7$ Again, we want to have the radical by itself. Let’s add $4$ on both sides. $3 \sqrt{x} - 4 + 4 = 7 + 4$ $3 \sqrt{x} = 11$ $\dfrac{3 \sqrt{x}}{3} = \dfrac{11}{3}$ $\sqrt{x} = \dfrac{11}{3}$ $(\sqrt{x})^2 = (\dfrac{11}{3})^2$ $x = \dfrac{121}{9}$ Let’s have this: $\sqrt{x - 8} = 3$ Here, the whole term is under the radical, so it is already isolated. So, we’ll start by squaring both sides. $(\sqrt{x - 8})^2 = 3^2$ $x - 8 = 9$ Now, let’s solve for $x$. $x - 8 + 8 = 9 + 8$ $x = 17$ # Part 2 In the second part, we are going to solve equations with a radical expression that are more complex. • Only$19.95/mo. No commitment. Cancel anytime.

For example:
$\sqrt{x}=(x-5)$

Square both sides of the equation
$\sqrt{x}^2=(x-5)^2$

Use FOIL to complete the squaring if necessary
$x=(x-5)(x-5)$
$x=x^2-5x-5x+25$

Combine like terms
$x=x^2-10x+25$

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$

Substitute the variables
$x=\frac{11\pm\sqrt{(-11)^2-4(1)(25)}}{2(1)}$
$x=\frac{11\pm\sqrt{121-4(1)(25)}}{2(1)}$
$x=\frac{11\pm\sqrt{21}}{2}$

So we have:
$x=\frac{11+\sqrt{21}}{2}$ and $x=\frac{11-\sqrt{21}}{2}$

In the end, x is approximately 7.8 or 3.2

If it was -7.8 instead of 7.8, then it cannot be a negative answer since the square root of a negative number is imaginary. In this case, the final answer would just be 3.2.

## Video-Lesson Transcript

This is how to solve equations with a radical expression part 2.

$\sqrt{x} = x - 5$

It’s a little bit different because we have $x$ on both sides.

Here, we want to get the radical by itself first before we combine the $x$ together.

In order to get rid of the radical, let’s square both sides.

$(\sqrt{x})^2 = (x - 5)^2$
$x = (x - 5) (x - 5)$
$x = x^2 - 5x - 5x + 25$
$x = x^2 - 10x + 25$

Now, we have a quadratic equation.

We want to get all the terms on one side so the other side be zero.

$x - x = x^2 - 10x - x + 25$
$0 = x^2 - 11x + 25$

This can not be factored so let’s just use the quadratic function to solve it.

Our formula is

$x = \dfrac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Here we have

$a = 1, b = -11, c = 25$
$x = \dfrac{11 \pm \sqrt{121 - 4(1)(25)}}{2 (1)}$
$x = \dfrac{11 \pm \sqrt{21}}{2}$

{$\dfrac{11 + \sqrt{21}}{2}, \dfrac{11 - \sqrt{21}}{2}$}

What is important here is that we always want to make sure that we won’t end up with a negative radical.

Let’s find out what is the exact value of this is

$\sqrt{21} = 4.6$
{$\dfrac{11 + 4.6}{2} = 7.8$}
{$\dfrac{11 - 4.6}{2} = 3.2$}

If one of the answers come out as a negative, that’s a rejected answer.

The only answer we accept is a positive number. Because you cannot find the square root of a negative number.

So, you always have to check and make sure that you don’t have a negative number under the radical.