Solve sqrt{11-x}=x+1

$$\sqrt { 11 - x } = x + 1$$

Solve for x

$$x=2$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{11-x}\right)^{2}=\left(x+1\right)^{2}$$

Calculate \sqrt{11-x} to the power of 2 and get 11-x.

$$11-x=\left(x+1\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+1\right)^{2}.

$$11-x=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$11-x-x^{2}=2x+1$$

Subtract 2x from both sides.

$$11-x-x^{2}-2x=1$$

Combine -x and -2x to get -3x.

$$11-3x-x^{2}=1$$

Subtract 1 from both sides.

$$11-3x-x^{2}-1=0$$

Subtract 1 from 11 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 negative, the negative number has greater absolute value than the positive. 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.

$$a=2 b=-5$$

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

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

Factor out x in the first and 5 in the second group.

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

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

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

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

$$x=2 x=-5$$

Substitute 2 for x in the equation \sqrt{11-x}=x+1.

$$\sqrt{11-2}=2+1$$

Simplify. The value x=2 satisfies the equation.

$$3=3$$

Substitute -5 for x in the equation \sqrt{11-x}=x+1.

$$\sqrt{11-\left(-5\right)}=-5+1$$

Simplify. The value x=-5 does not satisfy the equation because the left and the right hand side have opposite signs.

$$4=-4$$

Equation \sqrt{11-x}=x+1 has a unique solution.

$$x=2$$