9.6 Solve Equations with Square Roots

Before you get started, take this readiness quiz.

Simplify: ⓐ $\sqrt{9}$ ⓑ $9^{2}$.
If you missed this problem, review the reference and the reference.

Solve: $5 \left(x + 1\right) - 4 = 3 \left(2 x - 7\right)$.
If you missed this problem, review the reference.

Solve: $n^{2} - 6 n + 8 = 0$.
If you missed this problem, review the reference.

For the equation $\sqrt{x + 6} = x$:

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 3$ a solution?

For the equation $\sqrt{− x + 2} = x$:

ⓐ Is $x = −2$ a solution? ⓑ Is $x = 1$ a solution?

Solve: $\sqrt{3 x - 5} = 5$.

Solve: $\sqrt{4 x + 8} = 6$.

Solve: $\sqrt{3 m + 2} - 5 = 0$.

Solve: $\sqrt{10 z + 1} - 2 = 0$.

Solve: $\sqrt{3 p + 3} + 3 = 5$.

Solve: $\sqrt{5 q + 1} + 4 = 6$.

Solve: $\sqrt{2 r - 3} + 5 = 0$.

Solve: $\sqrt{7 s - 3} + 2 = 0$.

Solve: $\sqrt{x - 2} + 2 = x$.

Solve: $\sqrt{y - 5} + 5 = y$.

Solve: $\sqrt{m + 9} - m + 3 = 0$.

Solve: $\sqrt{n + 1} - n + 1 = 0$.

Solve: $2 \sqrt{4 a + 2} - 16 = 16$.

Solve: $3 \sqrt{6 b + 3} - 25 = 50$.

Solve: $\sqrt{4 z - 3} = \sqrt{3 z + 2}$.

Solve: $\sqrt{2 x - 5} = \sqrt{5 x + 3}$.

Solve: $\sqrt{7 y + 1} = \sqrt{2 y - 5}$.

Solve: $\sqrt{m} + 1 = \sqrt{m + 9}$.

Solve: $\sqrt{x} + 3 = \sqrt{x + 5}$.

Solve: $\sqrt{m} + 5 = \sqrt{m + 16}$.

Solve: $\sqrt{q - 2} + 3 = \sqrt{4 q + 1}$.

Solve: $\sqrt{y - 3} + 2 = \sqrt{4 y + 2}$.

Solve: $\sqrt{n - 4} + 5 = \sqrt{3 n + 3}$.

Katie wants to plant a square lawn in her front yard. She has enough sod to cover an area of 370 square feet. Use the formula $s = \sqrt{A}$ to find the length of each side of her lawn. Round your answer to the nearest tenth of a foot.

Sergio wants to make a square mosaic as an inlay for a table he is building. He has enough tile to cover an area of 2704 square centimeters. Use the formula $s = \sqrt{A}$ to find the length of each side of his mosaic. Round your answer to the nearest tenth of a centimeter.

A helicopter dropped a rescue package from a height of 1,296 feet. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the ground.

A window washer dropped a squeegee from a platform 196 feet above the sidewalk Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the squeegee to reach the sidewalk.

An accident investigator measured the skid marks of the car. The length of the skid marks was 76 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

The skid marks of a vehicle involved in an accident were 122 feet long. Use the formula $s = \sqrt{24 d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

Check whether the given values are solutions.

For the equation $\sqrt{x + 12} = x$: ⓐ Is $x = 4$ a solution? ⓑ Is $x = −3$ a solution?

Check whether the given values are solutions.

For the equation $\sqrt{t + 6} = t$: ⓐ Is $t = −2$ a solution? ⓑ Is $t = 3$ a solution?

Solve.

$\sqrt{5 y + 1} = 4$

Solve.

$\sqrt{5 x - 6} = 8$

Solve.

$\sqrt{2 m - 3} - 5 = 0$

Solve.

$\sqrt{6 v - 2} - 10 = 0$

Solve.

$\sqrt{5 q + 3} - 4 = 0$

Solve.

$\sqrt{6 n + 1} + 4 = 8$

Solve.

$\sqrt{5 v - 2} + 5 = 0$

Solve.

$\sqrt{2 m + 1} + 4 = 0$

Solve.

ⓐ $\sqrt{v - 10} + 10 = v$
ⓑ $\sqrt{y + 4} - y + 2 = 0$

Solve.

ⓐ $\sqrt{s - 8} - s = −8$
ⓑ $\sqrt{w + 25} - w + 5 = 0$

Solve.

$2 \sqrt{5 x + 1} - 8 = 0$

Solve.

$3 \sqrt{7 y + 1} - 10 = 8$

Solve.

$\sqrt{4 v + 3} = \sqrt{v - 6}$

Solve.

$\sqrt{12 c + 6} = \sqrt{10 - 4 c}$

Solve.

ⓐ $\sqrt{r} + 6 = \sqrt{r + 8}$
ⓑ $\sqrt{s - 3} + 2 = \sqrt{s + 4}$

Solve.

ⓐ $\sqrt{x} + 10 = \sqrt{x + 2}$
ⓑ $\sqrt{y - 2} + 2 = \sqrt{2 y + 4}$

Solve.

$\sqrt{8 u + 1} + 9 = 0$

Solve.

$\sqrt{d} - 2 = \sqrt{d - 20}$

Solve.

$\sqrt{9 p + 9} = \sqrt{10 p - 6}$

Solve. Round approximations to one decimal place.

Landscaping Vince wants to make a square patio in his yard. He has enough concrete to pave an area of 130 square feet. Use the formula $s = \sqrt{A}$ to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.

Solve. Round approximations to one decimal place.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the flare to reach the water.

Solve. Round approximations to one decimal place.

Gravity A construction worker dropped a hammer while building the Grand Canyon skywalk, 4000 feet above the Colorado River. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the hammer to reach the river.

Solve. Round approximations to one decimal place.

Accident investigation The skid marks for a car involved in an accident measured 216 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.

Solve. Round approximations to one decimal place.

Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was 117 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.

1. ⓐ Solve the equation $\sqrt{r + 4} - r + 2 = 0$.
2. ⓑ Explain why one of the “solutions” that was found was not actually a solution to the equation.