Solve sqrt{14-x}+2=x

$$\sqrt{ 14-x } +2=x$$

Solve for x

$$x=5$$

Solution Steps

Subtract 2 from both sides of the equation.

$$\sqrt{14-x}=x-2$$

Square both sides of the equation.

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

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

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

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

$$14-x=x^{2}-4x+4$$

Subtract x^{2} from both sides.

$$14-x-x^{2}=-4x+4$$

Add 4x to both sides.

$$14-x-x^{2}+4x=4$$

Combine -x and 4x to get 3x.

$$14+3x-x^{2}=4$$

Subtract 4 from both sides.

$$14+3x-x^{2}-4=0$$

Subtract 4 from 14 to get 10.

$$10+3x-x^{2}=0$$

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

$$-x^{2}+3x+10=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+10. To find a and b, set up a system to be solved.

$$a+b=3 ab=-10=-10$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -10.

$$-1,10 -2,5$$

Calculate the sum for each pair.

$$-1+10=9 -2+5=3$$

The solution is the pair that gives sum 3.

$$a=5 b=-2$$

Rewrite -x^{2}+3x+10 as \left(-x^{2}+5x\right)+\left(-2x+10\right).

$$\left(-x^{2}+5x\right)+\left(-2x+10\right)$$

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

$$-x\left(x-5\right)-2\left(x-5\right)$$

Factor out common term x-5 by using distributive property.

$$\left(x-5\right)\left(-x-2\right)$$

To find equation solutions, solve x-5=0 and -x-2=0.

$$x=5 x=-2$$

Substitute 5 for x in the equation \sqrt{14-x}+2=x.

$$\sqrt{14-5}+2=5$$

Simplify. The value x=5 satisfies the equation.

$$5=5$$

Substitute -2 for x in the equation \sqrt{14-x}+2=x.

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

Simplify. The value x=-2 does not satisfy the equation because the left and the right hand side have opposite signs.

$$6=-2$$

Equation \sqrt{14-x}=x-2 has a unique solution.

$$x=5$$