Solve for x square root of 2x-7=2x

$\sqrt{2 x - 7} = 2 x$

Step 1

To remove the radical on the left side of the equation, square both sides of the equation.

${\sqrt{2 x - 7}}^{2} = {(2 x)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x - 7}$ as ${(2 x - 7)}^{\frac{1}{2}}$ .

${({(2 x - 7)}^{\frac{1}{2}})}^{2} = {(2 x)}^{2}$

Simplify the left side.

Simplify ${({(2 x - 7)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x - 7)}^{\frac{1}{2}})}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

${(2 x - 7)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x - 7)}^{\frac{1}{2} \cdot 2} = {(2 x)}^{2}$

Rewrite the expression.

${(2 x - 7)}^{1} = {(2 x)}^{2}$

Simplify.

$2 x - 7 = {(2 x)}^{2}$

Simplify the right side.

Simplify ${(2 x)}^{2}$ .

Apply the product rule to $2 x$ .

$2 x - 7 = 2^{2} x^{2}$

Raise $2$ to the power of $2$ .

$2 x - 7 = 4 x^{2}$

Step 3

Solve for $x$ .

Subtract $4 x^{2}$ from both sides of the equation.

$2 x - 7 - 4 x^{2} = 0$

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = - 4$ , $b = 2$ , and $c = - 7$ into the quadratic formula and solve for $x$ .

$\frac{- 2 \pm \sqrt{2^{2} - 4 \cdot (- 4 \cdot - 7)}}{2 \cdot - 4}$

Simplify.

Simplify the numerator.

Raise $2$ to the power of $2$ .

$x = \frac{- 2 \pm \sqrt{4 - 4 \cdot - 4 \cdot - 7}}{2 \cdot - 4}$

Multiply $- 4 \cdot - 4 \cdot - 7$ .

Multiply $- 4$ by $- 4$ .

$x = \frac{- 2 \pm \sqrt{4 + 16 \cdot - 7}}{2 \cdot - 4}$

Multiply $16$ by $- 7$ .

$x = \frac{- 2 \pm \sqrt{4 - 112}}{2 \cdot - 4}$

Subtract $112$ from $4$ .

$x = \frac{- 2 \pm \sqrt{- 108}}{2 \cdot - 4}$

Rewrite $- 108$ as $- 1 (108)$ .

$x = \frac{- 2 \pm \sqrt{- 1 \cdot 108}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1 (108)}$ as $\sqrt{- 1} \cdot \sqrt{108}$ .

$x = \frac{- 2 \pm \sqrt{- 1} \cdot \sqrt{108}}{2 \cdot - 4}$

Rewrite $\sqrt{- 1}$ as $i$ .

$x = \frac{- 2 \pm i \cdot \sqrt{108}}{2 \cdot - 4}$

Rewrite $108$ as $6^{2} \cdot 3$ .

Factor $36$ out of $108$ .

$x = \frac{- 2 \pm i \cdot \sqrt{36 (3)}}{2 \cdot - 4}$

Rewrite $36$ as $6^{2}$ .

$x = \frac{- 2 \pm i \cdot \sqrt{6^{2} \cdot 3}}{2 \cdot - 4}$

Pull terms out from under the radical.

$x = \frac{- 2 \pm i \cdot (6 \sqrt{3})}{2 \cdot - 4}$

Move $6$ to the left of $i$ .

$x = \frac{- 2 \pm 6 i \sqrt{3}}{2 \cdot - 4}$

Multiply $2$ by $- 4$ .

$x = \frac{- 2 \pm 6 i \sqrt{3}}{- 8}$

Simplify $\frac{- 2 \pm 6 i \sqrt{3}}{- 8}$ .

$x = \frac{1 \pm 3 i \sqrt{3}}{4}$

The final answer is the combination of both solutions.

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

Solve for x square root of 2x-7=2x

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