Solve for x (2x-3)^2-8=0

${(2 x - 3)}^{2} - 8 = 0$

Step 1

Add $8$ to both sides of the equation.

${(2 x - 3)}^{2} = 8$

Step 2

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

$2 x - 3 = \pm \sqrt{8}$

Step 3

Simplify $\pm \sqrt{8}$ .

Rewrite $8$ as $2^{2} \cdot 2$ .

Factor $4$ out of $8$ .

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

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

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

Pull terms out from under the radical.

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

Step 4

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.

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

Add $3$ to both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

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

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

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

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

Add $3$ to both sides of the equation.

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

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Cancel the common factor of $- 2$ and $2$ .

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

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

Cancel the common factors.

Factor $2$ out of $2$ .

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

Cancel the common factor.

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

Rewrite the expression.

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

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

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

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

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.91421356 \dots, 0.08578643 \dots$

Solve for x (2x-3)^2-8=0

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