Solve sqrt{3x+22}=x+6

$$\sqrt { 3 x + 22 } = x + 6$$

Solve for x

$$x=-2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{3x+22}\right)^{2}=\left(x+6\right)^{2}$$

Calculate \sqrt{3x+22} to the power of 2 and get 3x+22.

$$3x+22=\left(x+6\right)^{2}$$

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

$$3x+22=x^{2}+12x+36$$

Subtract x^{2} from both sides.

$$3x+22-x^{2}=12x+36$$

Subtract 12x from both sides.

$$3x+22-x^{2}-12x=36$$

Combine 3x and -12x to get -9x.

$$-9x+22-x^{2}=36$$

Subtract 36 from both sides.

$$-9x+22-x^{2}-36=0$$

Subtract 36 from 22 to get -14.

$$-9x-14-x^{2}=0$$

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

$$-x^{2}-9x-14=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-14. To find a and b, set up a system to be solved.

$$a+b=-9 ab=-\left(-14\right)=14$$

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 14.

$$-1,-14 -2,-7$$

Calculate the sum for each pair.

$$-1-14=-15 -2-7=-9$$

The solution is the pair that gives sum -9.

$$a=-2 b=-7$$

Rewrite -x^{2}-9x-14 as \left(-x^{2}-2x\right)+\left(-7x-14\right).

$$\left(-x^{2}-2x\right)+\left(-7x-14\right)$$

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

$$x\left(-x-2\right)+7\left(-x-2\right)$$

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

$$\left(-x-2\right)\left(x+7\right)$$

To find equation solutions, solve -x-2=0 and x+7=0.

$$x=-2 x=-7$$

Substitute -2 for x in the equation \sqrt{3x+22}=x+6.

$$\sqrt{3\left(-2\right)+22}=-2+6$$

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

$$4=4$$

Substitute -7 for x in the equation \sqrt{3x+22}=x+6.

$$\sqrt{3\left(-7\right)+22}=-7+6$$

Simplify. The value x=-7 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{3x+22}=x+6 has a unique solution.

$$x=-2$$