Solve sqrt{x+14}=16-x

$$\sqrt{ x+14 } =16-x$$

Solve for x

$$x=11$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{x+14}\right)^{2}=\left(16-x\right)^{2}$$

Calculate \sqrt{x+14} to the power of 2 and get x+14.

$$x+14=\left(16-x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(16-x\right)^{2}.

$$x+14=256-32x+x^{2}$$

Subtract 256 from both sides.

$$x+14-256=-32x+x^{2}$$

Subtract 256 from 14 to get -242.

$$x-242=-32x+x^{2}$$

Add 32x to both sides.

$$x-242+32x=x^{2}$$

Combine x and 32x to get 33x.

$$33x-242=x^{2}$$

Subtract x^{2} from both sides.

$$33x-242-x^{2}=0$$

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

$$-x^{2}+33x-242=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx-242. To find a and b, set up a system to be solved.

$$a+b=33 ab=-\left(-242\right)=242$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 242.

$$1,242 2,121 11,22$$

Calculate the sum for each pair.

$$1+242=243 2+121=123 11+22=33$$

The solution is the pair that gives sum 33.

$$a=22 b=11$$

Rewrite -x^{2}+33x-242 as \left(-x^{2}+22x\right)+\left(11x-242\right).

$$\left(-x^{2}+22x\right)+\left(11x-242\right)$$

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

$$-x\left(x-22\right)+11\left(x-22\right)$$

Factor out common term x-22 by using distributive property.

$$\left(x-22\right)\left(-x+11\right)$$

To find equation solutions, solve x-22=0 and -x+11=0.

$$x=22 x=11$$

Substitute 22 for x in the equation \sqrt{x+14}=16-x.

$$\sqrt{22+14}=16-22$$

Simplify. The value x=22 does not satisfy the equation because the left and the right hand side have opposite signs.

$$6=-6$$

Substitute 11 for x in the equation \sqrt{x+14}=16-x.

$$\sqrt{11+14}=16-11$$

Simplify. The value x=11 satisfies the equation.

$$5=5$$

Equation \sqrt{x+14}=16-x has a unique solution.

$$x=11$$