Solve by Factoring 2x = square root of 20+2x

$2 x = \sqrt{20 + 2 x}$

Step 1

Subtract $\sqrt{20 + 2 x}$ from both sides of the equation.

$2 x - \sqrt{20 + 2 x} = 0$

Step 2

Factor $2$ out of $20 + 2 x$ .

Factor $2$ out of $20$ .

$2 x - \sqrt{2 \cdot 10 + 2 x} = 0$

Factor $2$ out of $2 \cdot 10 + 2 x$ .

$2 x - \sqrt{2 (10 + x)} = 0$

Step 3

Subtract $2 x$ from both sides of the equation.

$- \sqrt{2 (10 + x)} = - 2 x$

Step 4

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

${(- \sqrt{2 (10 + x)})}^{2} = {(- 2 x)}^{2}$

Step 5

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 (10 + x)}$ as ${(2 (10 + x))}^{\frac{1}{2}}$ .

${(- {(2 (10 + x))}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Simplify the left side.

Simplify ${(- {(2 (10 + x))}^{\frac{1}{2}})}^{2}$ .

Apply the distributive property.

${(- {(2 \cdot 10 + 2 x)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Simplify the expression.

Multiply $2$ by $10$ .

${(- {(20 + 2 x)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Apply the product rule to $- {(20 + 2 x)}^{\frac{1}{2}}$ .

${(- 1)}^{2} {({(20 + 2 x)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Raise $- 1$ to the power of $2$ .

$1 {({(20 + 2 x)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Multiply ${({(20 + 2 x)}^{\frac{1}{2}})}^{2}$ by $1$ .

${({(20 + 2 x)}^{\frac{1}{2}})}^{2} = {(- 2 x)}^{2}$

Multiply the exponents in ${({(20 + 2 x)}^{\frac{1}{2}})}^{2}$ .

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

${(20 + 2 x)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(20 + 2 x)}^{\frac{1}{2} \cdot 2} = {(- 2 x)}^{2}$

Rewrite the expression.

${(20 + 2 x)}^{1} = {(- 2 x)}^{2}$

Simplify.

$20 + 2 x = {(- 2 x)}^{2}$

Simplify the right side.

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

Apply the product rule to $- 2 x$ .

$20 + 2 x = {(- 2)}^{2} x^{2}$

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

$20 + 2 x = 4 x^{2}$

Step 6

Solve for $x$ .

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

$20 + 2 x - 4 x^{2} = 0$

Factor the left side of the equation.

Factor $- 2$ out of $20 + 2 x - 4 x^{2}$ .

Reorder the expression.

Move $20$ .

$2 x - 4 x^{2} + 20 = 0$

Reorder $2 x$ and $- 4 x^{2}$ .

$- 4 x^{2} + 2 x + 20 = 0$

Factor $- 2$ out of $- 4 x^{2}$ .

$- 2 (2 x^{2}) + 2 x + 20 = 0$

Factor $- 2$ out of $2 x$ .

$- 2 (2 x^{2}) - 2 (- x) + 20 = 0$

Factor $- 2$ out of $20$ .

$- 2 (2 x^{2}) - 2 (- x) - 2 \cdot - 10 = 0$

Factor $- 2$ out of $- 2 (2 x^{2}) - 2 (- x)$ .

$- 2 (2 x^{2} - x) - 2 \cdot - 10 = 0$

Factor $- 2$ out of $- 2 (2 x^{2} - x) - 2 (- 10)$ .

$- 2 (2 x^{2} - x - 10) = 0$

Factor.

Factor by grouping.

For a polynomial of the form $a x^{2} + b x + c$ , rewrite the middle term as a sum of two terms whose product is $a \cdot c = 2 \cdot - 10 = - 20$ and whose sum is $b = - 1$ .

Factor $- 1$ out of $- x$ .

$- 2 (2 x^{2} - (x) - 10) = 0$

Rewrite $- 1$ as $4$ plus $- 5$

$- 2 (2 x^{2} + (4 - 5) x - 10) = 0$

Apply the distributive property.

$- 2 (2 x^{2} + 4 x - 5 x - 10) = 0$

Factor out the greatest common factor from each group.

Group the first two terms and the last two terms.

$- 2 ((2 x^{2} + 4 x) - 5 x - 10) = 0$

Factor out the greatest common factor (GCF) from each group.

$- 2 (2 x (x + 2) - 5 (x + 2)) = 0$

Factor the polynomial by factoring out the greatest common factor, $x + 2$ .

$- 2 ((x + 2) (2 x - 5)) = 0$

Remove unnecessary parentheses.

$- 2 (x + 2) (2 x - 5) = 0$

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x + 2 = 0$

$2 x - 5 = 0$

Set $x + 2$ equal to $0$ and solve for $x$ .

Set $x + 2$ equal to $0$ .

$x + 2 = 0$

Subtract $2$ from both sides of the equation.

$x = - 2$

Set $2 x - 5$ equal to $0$ and solve for $x$ .

Set $2 x - 5$ equal to $0$ .

$2 x - 5 = 0$

Solve $2 x - 5 = 0$ for $x$ .

Add $5$ to both sides of the equation.

$2 x = 5$

Divide each term in $2 x = 5$ by $2$ and simplify.

Divide each term in $2 x = 5$ by $2$ .

$\frac{2 x}{2} = \frac{5}{2}$

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

$\frac{2 x}{2} = \frac{5}{2}$

Divide $x$ by $1$ .

$x = \frac{5}{2}$

The final solution is all the values that make $- 2 (x + 2) (2 x - 5) = 0$ true.

$x = - 2, \frac{5}{2}$

Step 7

Exclude the solutions that do not make $2 x - \sqrt{2 (10 + x)} = 0$ true.

$x = \frac{5}{2}$

Solve by Factoring 2x = square root of 20+2x

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