Solve Using the Square Root Property (8x+19)^2=45

${(8 x + 19)}^{2} = 45$

Step 1

Take the specified root of both sides of the equation to eliminate the exponent on the left side.

$8 x + 19 = \pm \sqrt{45}$

Step 2

Simplify $\pm \sqrt{45}$ .

Rewrite $45$ as $3^{2} \cdot 5$ .

Factor $9$ out of $45$ .

$8 x + 19 = \pm \sqrt{9 (5)}$

Rewrite $9$ as $3^{2}$ .

$8 x + 19 = \pm \sqrt{3^{2} \cdot 5}$

Pull terms out from under the radical.

$8 x + 19 = \pm 3 \sqrt{5}$

Step 3

The complete solution is the result of both the positive and negative portions of the solution.

First, use the positive value of the $\pm$ to find the first solution.

$8 x + 19 = 3 \sqrt{5}$

Subtract $19$ from both sides of the equation.

$8 x = 3 \sqrt{5} - 19$

Divide each term in $8 x = 3 \sqrt{5} - 19$ by $8$ and simplify.

Divide each term in $8 x = 3 \sqrt{5} - 19$ by $8$ .

$\frac{8 x}{8} = \frac{3 \sqrt{5}}{8} + \frac{- 19}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{3 \sqrt{5}}{8} + \frac{- 19}{8}$

Divide $x$ by $1$ .

$x = \frac{3 \sqrt{5}}{8} + \frac{- 19}{8}$

Simplify the right side.

Move the negative in front of the fraction.

$x = \frac{3 \sqrt{5}}{8} - \frac{19}{8}$

Next, use the negative value of the $\pm$ to find the second solution.

$8 x + 19 = - 3 \sqrt{5}$

Subtract $19$ from both sides of the equation.

$8 x = - 3 \sqrt{5} - 19$

Divide each term in $8 x = - 3 \sqrt{5} - 19$ by $8$ and simplify.

Divide each term in $8 x = - 3 \sqrt{5} - 19$ by $8$ .

$\frac{8 x}{8} = \frac{- 3 \sqrt{5}}{8} + \frac{- 19}{8}$

Simplify the left side.

Cancel the common factor of $8$ .

Cancel the common factor.

$\frac{8 x}{8} = \frac{- 3 \sqrt{5}}{8} + \frac{- 19}{8}$

Divide $x$ by $1$ .

$x = \frac{- 3 \sqrt{5}}{8} + \frac{- 19}{8}$

Simplify the right side.

Simplify each term.

Move the negative in front of the fraction.

$x = - \frac{3 \sqrt{5}}{8} + \frac{- 19}{8}$

Move the negative in front of the fraction.

$x = - \frac{3 \sqrt{5}}{8} - \frac{19}{8}$

The complete solution is the result of both the positive and negative portions of the solution.

$x = \frac{3 \sqrt{5}}{8} - \frac{19}{8}, - \frac{3 \sqrt{5}}{8} - \frac{19}{8}$

Step 4

The result can be shown in multiple forms.

Exact Form:

$x = \frac{3 \sqrt{5}}{8} - \frac{19}{8}, - \frac{3 \sqrt{5}}{8} - \frac{19}{8}$

Decimal Form:

$x = - 1.53647450 \dots, - 3.21352549 \dots$

Solve Using the Square Root Property (8x+19)^2=45

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