Solve for x square root of 3x^2+12x+13=2x+5

$\sqrt{3 x^{2} + 12 x + 13} = 2 x + 5$

Step 1

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

${\sqrt{3 x^{2} + 12 x + 13}}^{2} = {(2 x + 5)}^{2}$

Step 2

Simplify each side of the equation.

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{3 x^{2} + 12 x + 13}$ as ${(3 x^{2} + 12 x + 13)}^{\frac{1}{2}}$ .

${({(3 x^{2} + 12 x + 13)}^{\frac{1}{2}})}^{2} = {(2 x + 5)}^{2}$

Simplify the left side.

Simplify ${({(3 x^{2} + 12 x + 13)}^{\frac{1}{2}})}^{2}$ .

Multiply the exponents in ${({(3 x^{2} + 12 x + 13)}^{\frac{1}{2}})}^{2}$ .

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

${(3 x^{2} + 12 x + 13)}^{\frac{1}{2} \cdot 2} = {(2 x + 5)}^{2}$

Cancel the common factor of $2$ .

Cancel the common factor.

${(3 x^{2} + 12 x + 13)}^{\frac{1}{2} \cdot 2} = {(2 x + 5)}^{2}$

Rewrite the expression.

${(3 x^{2} + 12 x + 13)}^{1} = {(2 x + 5)}^{2}$

Simplify.

$3 x^{2} + 12 x + 13 = {(2 x + 5)}^{2}$

Simplify the right side.

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

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

$3 x^{2} + 12 x + 13 = (2 x + 5) (2 x + 5)$

Expand $(2 x + 5) (2 x + 5)$ using the FOIL Method.

Apply the distributive property.

$3 x^{2} + 12 x + 13 = 2 x (2 x + 5) + 5 (2 x + 5)$

Apply the distributive property.

$3 x^{2} + 12 x + 13 = 2 x (2 x) + 2 x \cdot 5 + 5 (2 x + 5)$

Apply the distributive property.

$3 x^{2} + 12 x + 13 = 2 x (2 x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5$

Simplify and combine like terms.

Simplify each term.

Rewrite using the commutative property of multiplication.

$3 x^{2} + 12 x + 13 = 2 \cdot 2 x \cdot x + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5$

Multiply $x$ by $x$ by adding the exponents.

Move $x$ .

$3 x^{2} + 12 x + 13 = 2 \cdot 2 (x \cdot x) + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5$

Multiply $x$ by $x$ .

$3 x^{2} + 12 x + 13 = 2 \cdot 2 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5$

Multiply $2$ by $2$ .

$3 x^{2} + 12 x + 13 = 4 x^{2} + 2 x \cdot 5 + 5 (2 x) + 5 \cdot 5$

Multiply $5$ by $2$ .

$3 x^{2} + 12 x + 13 = 4 x^{2} + 10 x + 5 (2 x) + 5 \cdot 5$

Multiply $2$ by $5$ .

$3 x^{2} + 12 x + 13 = 4 x^{2} + 10 x + 10 x + 5 \cdot 5$

Multiply $5$ by $5$ .

$3 x^{2} + 12 x + 13 = 4 x^{2} + 10 x + 10 x + 25$

Add $10 x$ and $10 x$ .

$3 x^{2} + 12 x + 13 = 4 x^{2} + 20 x + 25$

Step 3

Solve for $x$ .

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

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

$3 x^{2} + 12 x + 13 - 4 x^{2} = 20 x + 25$

Subtract $20 x$ from both sides of the equation.

$3 x^{2} + 12 x + 13 - 4 x^{2} - 20 x = 25$

Subtract $4 x^{2}$ from $3 x^{2}$ .

$- x^{2} + 12 x + 13 - 20 x = 25$

Subtract $20 x$ from $12 x$ .

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

Subtract $25$ from both sides of the equation.

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

Subtract $25$ from $13$ .

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

Factor the left side of the equation.

Factor $- 1$ out of $- x^{2} - 8 x - 12$ .

Factor $- 1$ out of $- x^{2}$ .

$- (x^{2}) - 8 x - 12 = 0$

Factor $- 1$ out of $- 8 x$ .

$- (x^{2}) - (8 x) - 12 = 0$

Rewrite $- 12$ as $- 1 (12)$ .

$- (x^{2}) - (8 x) - 1 \cdot 12 = 0$

Factor $- 1$ out of $- (x^{2}) - (8 x)$ .

$- (x^{2} + 8 x) - 1 \cdot 12 = 0$

Factor $- 1$ out of $- (x^{2} + 8 x) - 1 (12)$ .

$- (x^{2} + 8 x + 12) = 0$

Factor.

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.

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

Remove unnecessary parentheses.

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

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.

$x = - 2, - 6$

Step 4

Exclude the solutions that do not make $\sqrt{3 x^{2} + 12 x + 13} = 2 x + 5$ true.

$x = - 2$

Solve for x square root of 3x^2+12x+13=2x+5

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