Solve sqrt{2x+26}-9=x

$$\sqrt { 2 x + 26 } - 9 = x$$

Solve for x

$$x=-5$$

Solution Steps

Subtract -9 from both sides of the equation.

$$\sqrt{2x+26}=x+9$$

Square both sides of the equation.

$$\left(\sqrt{2x+26}\right)^{2}=\left(x+9\right)^{2}$$

Calculate \sqrt{2x+26} to the power of 2 and get 2x+26.

$$2x+26=\left(x+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+9\right)^{2}.

$$2x+26=x^{2}+18x+81$$

Subtract x^{2} from both sides.

$$2x+26-x^{2}=18x+81$$

Subtract 18x from both sides.

$$2x+26-x^{2}-18x=81$$

Combine 2x and -18x to get -16x.

$$-16x+26-x^{2}=81$$

Subtract 81 from both sides.

$$-16x+26-x^{2}-81=0$$

Subtract 81 from 26 to get -55.

$$-16x-55-x^{2}=0$$

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

$$-x^{2}-16x-55=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-55. To find a and b, set up a system to be solved.

$$a+b=-16 ab=-\left(-55\right)=55$$

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 55.

$$-1,-55 -5,-11$$

Calculate the sum for each pair.

$$-1-55=-56 -5-11=-16$$

The solution is the pair that gives sum -16.

$$a=-5 b=-11$$

Rewrite -x^{2}-16x-55 as \left(-x^{2}-5x\right)+\left(-11x-55\right).

$$\left(-x^{2}-5x\right)+\left(-11x-55\right)$$

Factor out x in the first and 11 in the second group.

$$x\left(-x-5\right)+11\left(-x-5\right)$$

Factor out common term -x-5 by using distributive property.

$$\left(-x-5\right)\left(x+11\right)$$

To find equation solutions, solve -x-5=0 and x+11=0.

$$x=-5 x=-11$$

Substitute -5 for x in the equation \sqrt{2x+26}-9=x.

$$\sqrt{2\left(-5\right)+26}-9=-5$$

Simplify. The value x=-5 satisfies the equation.

$$-5=-5$$

Substitute -11 for x in the equation \sqrt{2x+26}-9=x.

$$\sqrt{2\left(-11\right)+26}-9=-11$$

Simplify. The value x=-11 does not satisfy the equation.

$$-7=-11$$

Equation \sqrt{2x+26}=x+9 has a unique solution.

$$x=-5$$