Solutions Written as Square Roots

9.5.2 Solutions Written as Square Roots

9.5.2 • Solutions Written as Square Roots

Activity

Solve each equation. Use the notation when appropriate and express radicals in simplest form.

$x^{2} - 13 = - 12$

Compare your answer:

$x = \pm 1$

$(x - 6)^{2} = 0$

$x = 6$

$x^{2} + 9 = 0$

Compare your answer:

No real solution

So far in this activity you encountered radical solutions that were easy to simplify. Sometimes you will get radical solutions like $\sqrt{12}$ . Let’s review how to simplify more difficult radicals using perfect squares.

Step 1 - Look at the ways to factor the number under the radical.

For example, factor 12 from $\sqrt{12}$ .

$12 = 6 * 2$ and $12 = 4 * 3$

Step 2 - Choose the factorization of the number that includes a perfect square. Rewrite the number under the radical using that factorization.

$\sqrt{12} = \sqrt{4 * 3}$

Step 3 - Recall that $\sqrt{a * b} = \sqrt{a} * \sqrt{b}$ .

$\sqrt{4 * 3} = \sqrt{4} * \sqrt{3}$

Step 4 - Simplify the perfect squares.

$\sqrt{4} * \sqrt{3} = 2 \sqrt{3}$

So $\sqrt{12} = 2 \sqrt{3}$

$x^{2} = 18$

Compare your answer:

$x = \pm 3 \sqrt{2}$

$x^{2} + 1 = 18$

Compare your answer:

$x = \pm \sqrt{17}$

$(x + 1)^{2} = 18$

Compare your answer:

$x = - 1 \pm 3 \sqrt{2}$ (or equivalent)

Video: Solving Quadratics with Square Roots

Watch the following video to learn more about equations with solutions that are square roots.

Access multimedia content

Self Check

Solve

x^2-3=19

$x= \pm\; 22$
$x= \pm \sqrt{22}$
$x= \pm \sqrt{19}+3$
$x=-4$ , $x=4$

Additional Resources

Solving Quadratics with Radical Solutions

When solving quadratic equations, it is important to remember that:

Any positive number has two square roots, one positive and one negative, because there are two numbers that can be squared to make that number. (For example, $6^{2}$ and $(- 6)^{2}$ both equal 36, so 6 and -6 are both square roots of 36.)
The square root symbol $(\sqrt{})$ can be used to express the positive square root of a number. For example, the square root of 36 is 6, but it can also be written as $\sqrt{36}$ because $\sqrt{36} \cdot \sqrt{36} = 36$ .
To express the negative square root of a number, say 36, we can write -6 or $- \sqrt{36}$ .
When a number is not a perfect square (for example, 40), we can express its square roots by writing $\sqrt{40}$ and $- \sqrt{40}$ .

How could we write the solutions to an equation like $(x + 4)^{2} = 11$ ? This equation is saying, “something squared is 11.” To make the equation true, that something must be $\sqrt{11}$ or $- \sqrt{11}$ .

We can write:

$x + 4 = \sqrt{11}$ or $x + 4 = - \sqrt{11}$

$x = - 4 + \sqrt{11}$ or $x = - 4 + - \sqrt{11}$

A more compact way to write the two solutions to the equation is: $x = - 4 \pm \sqrt{11}$ .

Try it

Solving Quadratics with Radical Solutions

Solve $(x - 2)^{2} = 17$ .

Here is how to solve this quadratic equation:

Step 1 - Make sure the quadratic is isolated on one side.

$(x - 2)^{2} = 17$

Step 2 - Take the square root of both sides.

$\sqrt{(x - 2)^{2}} = \pm \sqrt{17}$

Step 3 - Simplify.

$x - 2 = \pm \sqrt{17}$

Step 4 - Solve the equation. For this problem, add 2 to each side.

$x - 2 + 2 = 2 \pm \sqrt{17}$

Step 5 - Simplify.

$x = 2 - \sqrt{17}$ , $x = 2 + \sqrt{17}$