Solve sqrt{(5x+11)}=x+1

$$\sqrt { ( 5 x + 11 ) } = x + 1$$

Solve for x

$$x=5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{5x+11}\right)^{2}=\left(x+1\right)^{2}$$

Calculate \sqrt{5x+11} to the power of 2 and get 5x+11.

$$5x+11=\left(x+1\right)^{2}$$

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

$$5x+11=x^{2}+2x+1$$

Subtract x^{2} from both sides.

$$5x+11-x^{2}=2x+1$$

Subtract 2x from both sides.

$$5x+11-x^{2}-2x=1$$

Combine 5x and -2x to get 3x.

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

Subtract 1 from both sides.

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

Subtract 1 from 11 to get 10.

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

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{5x+11}=x+1.

$$\sqrt{5\times 5+11}=5+1$$

Simplify. The value x=5 satisfies the equation.

$$6=6$$

Substitute -2 for x in the equation \sqrt{5x+11}=x+1.

$$\sqrt{5\left(-2\right)+11}=-2+1$$

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

$$1=-1$$

Equation \sqrt{5x+11}=x+1 has a unique solution.

$$x=5$$