Solve sqrt{2x+13}-3x=9

$$\sqrt { 2 x + 13 } - 3 x = 9$$

Solve for x

$$x=-2$$

Solution Steps

Subtract -3x from both sides of the equation.

$$\sqrt{2x+13}=9+3x$$

Square both sides of the equation.

$$\left(\sqrt{2x+13}\right)^{2}=\left(9+3x\right)^{2}$$

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

$$2x+13=\left(9+3x\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(9+3x\right)^{2}.

$$2x+13=81+54x+9x^{2}$$

Subtract 81 from both sides.

$$2x+13-81=54x+9x^{2}$$

Subtract 81 from 13 to get -68.

$$2x-68=54x+9x^{2}$$

Subtract 54x from both sides.

$$2x-68-54x=9x^{2}$$

Combine 2x and -54x to get -52x.

$$-52x-68=9x^{2}$$

Subtract 9x^{2} from both sides.

$$-52x-68-9x^{2}=0$$

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

$$-9x^{2}-52x-68=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-68. To find a and b, set up a system to be solved.

$$a+b=-52 ab=-9\left(-68\right)=612$$

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 612.

$$-1,-612 -2,-306 -3,-204 -4,-153 -6,-102 -9,-68 -12,-51 -17,-36 -18,-34$$

Calculate the sum for each pair.

$$-1-612=-613 -2-306=-308 -3-204=-207 -4-153=-157 -6-102=-108 -9-68=-77 -12-51=-63 -17-36=-53 -18-34=-52$$

The solution is the pair that gives sum -52.

$$a=-18 b=-34$$

Rewrite -9x^{2}-52x-68 as \left(-9x^{2}-18x\right)+\left(-34x-68\right).

$$\left(-9x^{2}-18x\right)+\left(-34x-68\right)$$

Factor out 9x in the first and 34 in the second group.

$$9x\left(-x-2\right)+34\left(-x-2\right)$$

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

$$\left(-x-2\right)\left(9x+34\right)$$

To find equation solutions, solve -x-2=0 and 9x+34=0.

$$x=-2 x=-\frac{34}{9}$$

Substitute -2 for x in the equation \sqrt{2x+13}-3x=9.

$$\sqrt{2\left(-2\right)+13}-3\left(-2\right)=9$$

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

$$9=9$$

Substitute -\frac{34}{9} for x in the equation \sqrt{2x+13}-3x=9.

$$\sqrt{2\left(-\frac{34}{9}\right)+13}-3\left(-\frac{34}{9}\right)=9$$

Simplify. The value x=-\frac{34}{9} does not satisfy the equation.

$$\frac{41}{3}=9$$

Equation \sqrt{2x+13}=3x+9 has a unique solution.

$$x=-2$$