Solve sqrt{y+4}-4=y

$$\sqrt { y + 4 } - 4 = y$$

Solve for y

$$y=-3$$

$$y=-4$$

Solution Steps

Subtract -4 from both sides of the equation.

$$\sqrt{y+4}=y+4$$

Square both sides of the equation.

$$\left(\sqrt{y+4}\right)^{2}=\left(y+4\right)^{2}$$

Calculate \sqrt{y+4} to the power of 2 and get y+4.

$$y+4=\left(y+4\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(y+4\right)^{2}.

$$y+4=y^{2}+8y+16$$

Subtract y^{2} from both sides.

$$y+4-y^{2}=8y+16$$

Subtract 8y from both sides.

$$y+4-y^{2}-8y=16$$

Combine y and -8y to get -7y.

$$-7y+4-y^{2}=16$$

Subtract 16 from both sides.

$$-7y+4-y^{2}-16=0$$

Subtract 16 from 4 to get -12.

$$-7y-12-y^{2}=0$$

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

$$-y^{2}-7y-12=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -y^{2}+ay+by-12. To find a and b, set up a system to be solved.

$$a+b=-7 ab=-\left(-12\right)=12$$

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 12.

$$-1,-12 -2,-6 -3,-4$$

Calculate the sum for each pair.

$$-1-12=-13 -2-6=-8 -3-4=-7$$

The solution is the pair that gives sum -7.

$$a=-3 b=-4$$

Rewrite -y^{2}-7y-12 as \left(-y^{2}-3y\right)+\left(-4y-12\right).

$$\left(-y^{2}-3y\right)+\left(-4y-12\right)$$

Factor out y in the first and 4 in the second group.

$$y\left(-y-3\right)+4\left(-y-3\right)$$

Factor out common term -y-3 by using distributive property.

$$\left(-y-3\right)\left(y+4\right)$$

To find equation solutions, solve -y-3=0 and y+4=0.

$$y=-3 y=-4$$

Substitute -3 for y in the equation \sqrt{y+4}-4=y.

$$\sqrt{-3+4}-4=-3$$

Simplify. The value y=-3 satisfies the equation.

$$-3=-3$$

Substitute -4 for y in the equation \sqrt{y+4}-4=y.

$$\sqrt{-4+4}-4=-4$$

Simplify. The value y=-4 satisfies the equation.

$$-4=-4$$

List all solutions of \sqrt{y+4}=y+4.

$$y=-3 y=-4$$