Solve sqrt{9x+55}=x+5

$$\sqrt { 9 x + 55 } = x + 5$$

Solve for x

$$x=5$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9x+55}\right)^{2}=\left(x+5\right)^{2}$$

Calculate \sqrt{9x+55} to the power of 2 and get 9x+55.

$$9x+55=\left(x+5\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+5\right)^{2}.

$$9x+55=x^{2}+10x+25$$

Subtract x^{2} from both sides.

$$9x+55-x^{2}=10x+25$$

Subtract 10x from both sides.

$$9x+55-x^{2}-10x=25$$

Combine 9x and -10x to get -x.

$$-x+55-x^{2}=25$$

Subtract 25 from both sides.

$$-x+55-x^{2}-25=0$$

Subtract 25 from 55 to get 30.

$$-x+30-x^{2}=0$$

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

$$-x^{2}-x+30=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=-1 ab=-30=-30$$

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 -30.

$$1,-30 2,-15 3,-10 5,-6$$

Calculate the sum for each pair.

$$1-30=-29 2-15=-13 3-10=-7 5-6=-1$$

The solution is the pair that gives sum -1.

$$a=5 b=-6$$

Rewrite -x^{2}-x+30 as \left(-x^{2}+5x\right)+\left(-6x+30\right).

$$\left(-x^{2}+5x\right)+\left(-6x+30\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(-x+5\right)+6\left(-x+5\right)$$

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

$$\left(-x+5\right)\left(x+6\right)$$

To find equation solutions, solve -x+5=0 and x+6=0.

$$x=5 x=-6$$

Substitute 5 for x in the equation \sqrt{9x+55}=x+5.

$$\sqrt{9\times 5+55}=5+5$$

Simplify. The value x=5 satisfies the equation.

$$10=10$$

Substitute -6 for x in the equation \sqrt{9x+55}=x+5.

$$\sqrt{9\left(-6\right)+55}=-6+5$$

Simplify. The value x=-6 does not satisfy the equation because the left and the right hand side have opposite signs.

$$1=-1$$

Equation \sqrt{9x+55}=x+5 has a unique solution.

$$x=5$$