Solve 3x+sqrt{4x+9}=-5

$$3x+ \sqrt{ 4x+9 } = -5$$

Solve for x

$$x=-2$$

Solution Steps

Subtract 3x from both sides of the equation.

$$\sqrt{4x+9}=-5-3x$$

Square both sides of the equation.

$$\left(\sqrt{4x+9}\right)^{2}=\left(-5-3x\right)^{2}$$

Calculate \sqrt{4x+9} to the power of 2 and get 4x+9.

$$4x+9=\left(-5-3x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(-5-3x\right)^{2}.

$$4x+9=25+30x+9x^{2}$$

Subtract 25 from both sides.

$$4x+9-25=30x+9x^{2}$$

Subtract 25 from 9 to get -16.

$$4x-16=30x+9x^{2}$$

Subtract 30x from both sides.

$$4x-16-30x=9x^{2}$$

Combine 4x and -30x to get -26x.

$$-26x-16=9x^{2}$$

Subtract 9x^{2} from both sides.

$$-26x-16-9x^{2}=0$$

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

$$-9x^{2}-26x-16=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -9x^{2}+ax+bx-16. To find a and b, set up a system to be solved.

$$a+b=-26 ab=-9\left(-16\right)=144$$

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 144.

$$-1,-144 -2,-72 -3,-48 -4,-36 -6,-24 -8,-18 -9,-16 -12,-12$$

Calculate the sum for each pair.

$$-1-144=-145 -2-72=-74 -3-48=-51 -4-36=-40 -6-24=-30 -8-18=-26 -9-16=-25 -12-12=-24$$

The solution is the pair that gives sum -26.

$$a=-8 b=-18$$

Rewrite -9x^{2}-26x-16 as \left(-9x^{2}-8x\right)+\left(-18x-16\right).

$$\left(-9x^{2}-8x\right)+\left(-18x-16\right)$$

Factor out -x in the first and -2 in the second group.

$$-x\left(9x+8\right)-2\left(9x+8\right)$$

Factor out common term 9x+8 by using distributive property.

$$\left(9x+8\right)\left(-x-2\right)$$

To find equation solutions, solve 9x+8=0 and -x-2=0.

$$x=-\frac{8}{9} x=-2$$

Substitute -\frac{8}{9} for x in the equation 3x+\sqrt{4x+9}=-5.

$$3\left(-\frac{8}{9}\right)+\sqrt{4\left(-\frac{8}{9}\right)+9}=-5$$

Simplify. The value x=-\frac{8}{9} does not satisfy the equation.

$$-\frac{1}{3}=-5$$

Substitute -2 for x in the equation 3x+\sqrt{4x+9}=-5.

$$3\left(-2\right)+\sqrt{4\left(-2\right)+9}=-5$$

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

$$-5=-5$$

Equation \sqrt{4x+9}=-3x-5 has a unique solution.

$$x=-2$$