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

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

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

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

Calculate \sqrt{3x^{2}+12x+13} to the power of 2 and get 3x^{2}+12x+13.

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2x+5\right)^{2}.

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

Subtract 4x^{2} from both sides.

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

Combine 3x^{2} and -4x^{2} to get -x^{2}.

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

Subtract 20x from both sides.

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

Combine 12x and -20x to get -8x.

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

Subtract 25 from both sides.

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

Subtract 25 from 13 to get -12.

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

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-12. To find a and b, set up a system to be solved.

$$a+b=-8 ab=-\left(-12\right)=12$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -8.

$$a=-2 b=-6$$

Rewrite -x^{2}-8x-12 as \left(-x^{2}-2x\right)+\left(-6x-12\right).

$$\left(-x^{2}-2x\right)+\left(-6x-12\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x-2\right)+6\left(-x-2\right)$$

Factor out common term -x-2 by using distributive property.

$$\left(-x-2\right)\left(x+6\right)$$

To find equation solutions, solve -x-2=0 and x+6=0.

$$x=-2 x=-6$$

Substitute -2 for x in the equation \sqrt{3x^{2}+12x+13}=2x+5.

$$\sqrt{3\left(-2\right)^{2}+12\left(-2\right)+13}=2\left(-2\right)+5$$

Simplify. The value x=-2 satisfies the equation.

$$1=1$$

Substitute -6 for x in the equation \sqrt{3x^{2}+12x+13}=2x+5.

$$\sqrt{3\left(-6\right)^{2}+12\left(-6\right)+13}=2\left(-6\right)+5$$

Simplify. The value x=-6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$7=-7$$

Equation \sqrt{3x^{2}+12x+13}=2x+5 has a unique solution.

$$x=-2$$