Solve sqrt{2x+22}=x+7

$$\sqrt { 2 x + 22 } = x + 7$$

Solve for x

$$x=-3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2x+22}\right)^{2}=\left(x+7\right)^{2}$$

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

$$2x+22=\left(x+7\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$2x+22=x^{2}+14x+49$$

Subtract x^{2} from both sides.

$$2x+22-x^{2}=14x+49$$

Subtract 14x from both sides.

$$2x+22-x^{2}-14x=49$$

Combine 2x and -14x to get -12x.

$$-12x+22-x^{2}=49$$

Subtract 49 from both sides.

$$-12x+22-x^{2}-49=0$$

Subtract 49 from 22 to get -27.

$$-12x-27-x^{2}=0$$

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

$$-x^{2}-12x-27=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-27. To find a and b, set up a system to be solved.

$$a+b=-12 ab=-\left(-27\right)=27$$

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 27.

$$-1,-27 -3,-9$$

Calculate the sum for each pair.

$$-1-27=-28 -3-9=-12$$

The solution is the pair that gives sum -12.

$$a=-3 b=-9$$

Rewrite -x^{2}-12x-27 as \left(-x^{2}-3x\right)+\left(-9x-27\right).

$$\left(-x^{2}-3x\right)+\left(-9x-27\right)$$

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

$$x\left(-x-3\right)+9\left(-x-3\right)$$

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

$$\left(-x-3\right)\left(x+9\right)$$

To find equation solutions, solve -x-3=0 and x+9=0.

$$x=-3 x=-9$$

Substitute -3 for x in the equation \sqrt{2x+22}=x+7.

$$\sqrt{2\left(-3\right)+22}=-3+7$$

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

$$4=4$$

Substitute -9 for x in the equation \sqrt{2x+22}=x+7.

$$\sqrt{2\left(-9\right)+22}=-9+7$$

Simplify. The value x=-9 does not satisfy the equation because the left and the right hand side have opposite signs.

$$2=-2$$

Equation \sqrt{2x+22}=x+7 has a unique solution.

$$x=-3$$