Solve sqrt{2y+17}=y+9

$$\sqrt { 2 y + 17 } = y + 9$$

Solve for y

$$y=-8$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{2y+17}\right)^{2}=\left(y+9\right)^{2}$$

Calculate \sqrt{2y+17} to the power of 2 and get 2y+17.

$$2y+17=\left(y+9\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+9\right)^{2}.

$$2y+17=y^{2}+18y+81$$

Subtract y^{2} from both sides.

$$2y+17-y^{2}=18y+81$$

Subtract 18y from both sides.

$$2y+17-y^{2}-18y=81$$

Combine 2y and -18y to get -16y.

$$-16y+17-y^{2}=81$$

Subtract 81 from both sides.

$$-16y+17-y^{2}-81=0$$

Subtract 81 from 17 to get -64.

$$-16y-64-y^{2}=0$$

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

$$-y^{2}-16y-64=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-64. To find a and b, set up a system to be solved.

$$a+b=-16 ab=-\left(-64\right)=64$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 64.

$$-1,-64 -2,-32 -4,-16 -8,-8$$

Calculate the sum for each pair.

$$-1-64=-65 -2-32=-34 -4-16=-20 -8-8=-16$$

The solution is the pair that gives sum -16.

$$a=-8 b=-8$$

Rewrite -y^{2}-16y-64 as \left(-y^{2}-8y\right)+\left(-8y-64\right).

$$\left(-y^{2}-8y\right)+\left(-8y-64\right)$$

Factor out y in the first and 8 in the second group.

$$y\left(-y-8\right)+8\left(-y-8\right)$$

Factor out common term -y-8 by using distributive property.

$$\left(-y-8\right)\left(y+8\right)$$

To find equation solutions, solve -y-8=0 and y+8=0.

$$y=-8 y=-8$$

Substitute -8 for y in the equation \sqrt{2y+17}=y+9.

$$\sqrt{2\left(-8\right)+17}=-8+9$$

Simplify. The value y=-8 satisfies the equation.

$$1=1$$

Substitute -8 for y in the equation \sqrt{2y+17}=y+9.

$$\sqrt{2\left(-8\right)+17}=-8+9$$

Simplify. The value y=-8 satisfies the equation.

$$1=1$$

List all solutions of \sqrt{2y+17}=y+9.

$$y=-8 y=-8$$