Solve Using the Square Root Property (4x-3)^2+7=39

${(4 x - 3)}^{2} + 7 = 39$

Step 1

Subtract $7$ from both sides of the equation.

${(4 x - 3)}^{2} = 39 - 7$

Step 2

Subtract $7$ from $39$ .

${(4 x - 3)}^{2} = 32$

Step 3

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

$4 x - 3 = \pm \sqrt{32}$

Step 4

Simplify $\pm \sqrt{32}$ .

Rewrite $32$ as $4^{2} \cdot 2$ .

Factor $16$ out of $32$ .

$4 x - 3 = \pm \sqrt{16 (2)}$

Rewrite $16$ as $4^{2}$ .

$4 x - 3 = \pm \sqrt{4^{2} \cdot 2}$

Pull terms out from under the radical.

$4 x - 3 = \pm 4 \sqrt{2}$

Step 5

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.

$4 x - 3 = 4 \sqrt{2}$

Add $3$ to both sides of the equation.

$4 x = 4 \sqrt{2} + 3$

Divide each term in $4 x = 4 \sqrt{2} + 3$ by $4$ and simplify.

Divide each term in $4 x = 4 \sqrt{2} + 3$ by $4$ .

$\frac{4 x}{4} = \frac{4 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{4 \sqrt{2}}{4} + \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{4 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the right side.

Cancel the common factor of $4$ .

Cancel the common factor.

$x = \frac{4 \sqrt{2}}{4} + \frac{3}{4}$

Divide $\sqrt{2}$ by $1$ .

$x = \sqrt{2} + \frac{3}{4}$

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

$4 x - 3 = - 4 \sqrt{2}$

Add $3$ to both sides of the equation.

$4 x = - 4 \sqrt{2} + 3$

Divide each term in $4 x = - 4 \sqrt{2} + 3$ by $4$ and simplify.

Divide each term in $4 x = - 4 \sqrt{2} + 3$ by $4$ .

$\frac{4 x}{4} = \frac{- 4 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the left side.

Cancel the common factor of $4$ .

Cancel the common factor.

$\frac{4 x}{4} = \frac{- 4 \sqrt{2}}{4} + \frac{3}{4}$

Divide $x$ by $1$ .

$x = \frac{- 4 \sqrt{2}}{4} + \frac{3}{4}$

Simplify the right side.

Cancel the common factor of $- 4$ and $4$ .

Factor $4$ out of $- 4 \sqrt{2}$ .

$x = \frac{4 (- \sqrt{2})}{4} + \frac{3}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x = \frac{4 (- \sqrt{2})}{4 (1)} + \frac{3}{4}$

Cancel the common factor.

$x = \frac{4 (- \sqrt{2})}{4 \cdot 1} + \frac{3}{4}$

Rewrite the expression.

$x = \frac{- \sqrt{2}}{1} + \frac{3}{4}$

Divide $- \sqrt{2}$ by $1$ .

$x = - \sqrt{2} + \frac{3}{4}$

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

$x = \sqrt{2} + \frac{3}{4}, - \sqrt{2} + \frac{3}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \sqrt{2} + \frac{3}{4}, - \sqrt{2} + \frac{3}{4}$

Decimal Form:

$x = 2.16421356 \dots, - 0.66421356 \dots$

Solve Using the Square Root Property (4x-3)^2+7=39

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