Solve sqrt{2x+15}-6=x

$$\sqrt { 2 x + 15 } - 6 = x$$

Solve for x

$$x=-3$$

Solution Steps

Subtract -6 from both sides of the equation.

$$\sqrt{2x+15}=x+6$$

Square both sides of the equation.

$$\left(\sqrt{2x+15}\right)^{2}=\left(x+6\right)^{2}$$

Calculate \sqrt{2x+15} to the power of 2 and get 2x+15.

$$2x+15=\left(x+6\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+6\right)^{2}.

$$2x+15=x^{2}+12x+36$$

Subtract x^{2} from both sides.

$$2x+15-x^{2}=12x+36$$

Subtract 12x from both sides.

$$2x+15-x^{2}-12x=36$$

Combine 2x and -12x to get -10x.

$$-10x+15-x^{2}=36$$

Subtract 36 from both sides.

$$-10x+15-x^{2}-36=0$$

Subtract 36 from 15 to get -21.

$$-10x-21-x^{2}=0$$

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

$$-x^{2}-10x-21=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=-10 ab=-\left(-21\right)=21$$

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 21.

$$-1,-21 -3,-7$$

Calculate the sum for each pair.

$$-1-21=-22 -3-7=-10$$

The solution is the pair that gives sum -10.

$$a=-3 b=-7$$

Rewrite -x^{2}-10x-21 as \left(-x^{2}-3x\right)+\left(-7x-21\right).

$$\left(-x^{2}-3x\right)+\left(-7x-21\right)$$

Factor out x in the first and 7 in the second group.

$$x\left(-x-3\right)+7\left(-x-3\right)$$

Factor out common term -x-3 by using distributive property.

$$\left(-x-3\right)\left(x+7\right)$$

To find equation solutions, solve -x-3=0 and x+7=0.

$$x=-3 x=-7$$

Substitute -3 for x in the equation \sqrt{2x+15}-6=x.

$$\sqrt{2\left(-3\right)+15}-6=-3$$

Simplify. The value x=-3 satisfies the equation.

$$-3=-3$$

Substitute -7 for x in the equation \sqrt{2x+15}-6=x.

$$\sqrt{2\left(-7\right)+15}-6=-7$$

Simplify. The value x=-7 does not satisfy the equation.

$$-5=-7$$

Equation \sqrt{2x+15}=x+6 has a unique solution.

$$x=-3$$