Find All Complex Solutions x+5 = square root of 2x+13

$x + 5 = \sqrt{2 x + 13}$

Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

$\sqrt{2 x + 13} = x + 5$

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

${\sqrt{2 x + 13}}^{2} = {(x + 5)}^{2}$

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{2 x + 13}$ as ${(2 x + 13)}^{\frac{1}{2}}$ .

${({(2 x + 13)}^{\frac{1}{2}})}^{2} = {(x + 5)}^{2}$

Simplify the left side.

Simplify ${({(2 x + 13)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(2 x + 13)}^{\frac{1}{2}})}^{2}$ .

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

${(2 x + 13)}^{\frac{1}{2} \cdot 2} = {(x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(2 x + 13)}^{\frac{1}{2} \cdot 2} = {(x + 5)}^{2}$

Rewrite the expression.

${(2 x + 13)}^{1} = {(x + 5)}^{2}$

Simplify.

$2 x + 13 = {(x + 5)}^{2}$

Simplify the right side.

Simplify ${(x + 5)}^{2}$ .

Rewrite ${(x + 5)}^{2}$ as $(x + 5) (x + 5)$ .

$2 x + 13 = (x + 5) (x + 5)$

Expand $(x + 5) (x + 5)$ using the FOIL Method.

Apply the distributive property.

$2 x + 13 = x (x + 5) + 5 (x + 5)$

Apply the distributive property.

$2 x + 13 = x \cdot x + x \cdot 5 + 5 (x + 5)$

Apply the distributive property.

$2 x + 13 = x \cdot x + x \cdot 5 + 5 x + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$2 x + 13 = x^{2} + x \cdot 5 + 5 x + 5 \cdot 5$

Move $5$ to the left of $x$ .

$2 x + 13 = x^{2} + 5 \cdot x + 5 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$2 x + 13 = x^{2} + 5 x + 5 x + 25$

Add $5 x$ and $5 x$ .

$2 x + 13 = x^{2} + 10 x + 25$

Solve for $x$ .

Since $x$ is on the right side of the equation, switch the sides so it is on the left side of the equation.

$x^{2} + 10 x + 25 = 2 x + 13$

Move all terms containing $x$ to the left side of the equation.

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

$x^{2} + 10 x + 25 - 2 x = 13$

Subtract $2 x$ from $10 x$ .

$x^{2} + 8 x + 25 = 13$

Subtract $13$ from both sides of the equation.

$x^{2} + 8 x + 25 - 13 = 0$

Subtract $13$ from $25$ .

$x^{2} + 8 x + 12 = 0$

Factor $x^{2} + 8 x + 12$ using the AC method.

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $12$ and whose sum is $8$ .

$2, 6$

Write the factored form using these integers.

$(x + 2) (x + 6) = 0$

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

$x + 2 = 0$

$x + 6 = 0$

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 $x + 6$ equal to $0$ and solve for $x$ .

Set $x + 6$ equal to $0$ .

$x + 6 = 0$

Subtract $6$ from both sides of the equation.

$x = - 6$

The final solution is all the values that make $(x + 2) (x + 6) = 0$ true.

$x = - 2, - 6$

Exclude the solutions that do not make $x + 5 = \sqrt{2 x + 13}$ true.

$x = - 2$

Find All Complex Solutions x+5 = square root of 2x+13

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