Solve sqrt{2x+13}=x+5

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

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

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

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

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

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

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

Subtract x^{2} from both sides.

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

Subtract 10x from both sides.

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

Combine 2x and -10x to get -8x.

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

Subtract 25 from both sides.

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

Subtract 25 from 13 to get -12.

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

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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{2x+13}=x+5.

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

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

$$3=3$$

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

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

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

$$1=-1$$

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

$$x=-2$$