Solve sqrt{4y+20}-sqrt{y-4}=6

$$\sqrt { 4 y + 20 } - \sqrt { y - 4 } = 6$$

Solve for y

$$y=20$$

$$y=4$$

Solution Steps

Subtract -\sqrt{y-4} from both sides of the equation.

$$\sqrt{4y+20}=6+\sqrt{y-4}$$

Square both sides of the equation.

$$\left(\sqrt{4y+20}\right)^{2}=\left(6+\sqrt{y-4}\right)^{2}$$

Calculate \sqrt{4y+20} to the power of 2 and get 4y+20.

$$4y+20=\left(6+\sqrt{y-4}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(6+\sqrt{y-4}\right)^{2}.

$$4y+20=36+12\sqrt{y-4}+\left(\sqrt{y-4}\right)^{2}$$

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

$$4y+20=36+12\sqrt{y-4}+y-4$$

Subtract 4 from 36 to get 32.

$$4y+20=32+12\sqrt{y-4}+y$$

Subtract 32+y from both sides of the equation.

$$4y+20-\left(32+y\right)=12\sqrt{y-4}$$

To find the opposite of 32+y, find the opposite of each term.

$$4y+20-32-y=12\sqrt{y-4}$$

Subtract 32 from 20 to get -12.

$$4y-12-y=12\sqrt{y-4}$$

Combine 4y and -y to get 3y.

$$3y-12=12\sqrt{y-4}$$

Square both sides of the equation.

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

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3y-12\right)^{2}.

$$9y^{2}-72y+144=\left(12\sqrt{y-4}\right)^{2}$$

Expand \left(12\sqrt{y-4}\right)^{2}.

$$9y^{2}-72y+144=12^{2}\left(\sqrt{y-4}\right)^{2}$$

Calculate 12 to the power of 2 and get 144.

$$9y^{2}-72y+144=144\left(\sqrt{y-4}\right)^{2}$$

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

$$9y^{2}-72y+144=144\left(y-4\right)$$

Use the distributive property to multiply 144 by y-4.

$$9y^{2}-72y+144=144y-576$$

Subtract 144y from both sides.

$$9y^{2}-72y+144-144y=-576$$

Combine -72y and -144y to get -216y.

$$9y^{2}-216y+144=-576$$

Add 576 to both sides.

$$9y^{2}-216y+144+576=0$$

Add 144 and 576 to get 720.

$$9y^{2}-216y+720=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -216 for b, and 720 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-216\right)±\sqrt{\left(-216\right)^{2}-4\times 9\times 720}}{2\times 9}$$

Square -216.

$$y=\frac{-\left(-216\right)±\sqrt{46656-4\times 9\times 720}}{2\times 9}$$

Multiply -4 times 9.

$$y=\frac{-\left(-216\right)±\sqrt{46656-36\times 720}}{2\times 9}$$

Multiply -36 times 720.

$$y=\frac{-\left(-216\right)±\sqrt{46656-25920}}{2\times 9}$$

Add 46656 to -25920.

$$y=\frac{-\left(-216\right)±\sqrt{20736}}{2\times 9}$$

Take the square root of 20736.

$$y=\frac{-\left(-216\right)±144}{2\times 9}$$

The opposite of -216 is 216.

$$y=\frac{216±144}{2\times 9}$$

Multiply 2 times 9.

$$y=\frac{216±144}{18}$$

Now solve the equation y=\frac{216±144}{18} when ± is plus. Add 216 to 144.

$$y=\frac{360}{18}$$

Divide 360 by 18.

$$y=20$$

Now solve the equation y=\frac{216±144}{18} when ± is minus. Subtract 144 from 216.

$$y=\frac{72}{18}$$

Divide 72 by 18.

$$y=4$$

The equation is now solved.

$$y=20 y=4$$

Substitute 20 for y in the equation \sqrt{4y+20}-\sqrt{y-4}=6.

$$\sqrt{4\times 20+20}-\sqrt{20-4}=6$$

Simplify. The value y=20 satisfies the equation.

$$6=6$$

Substitute 4 for y in the equation \sqrt{4y+20}-\sqrt{y-4}=6.

$$\sqrt{4\times 4+20}-\sqrt{4-4}=6$$

Simplify. The value y=4 satisfies the equation.

$$6=6$$

List all solutions of \sqrt{4y+20}=\sqrt{y-4}+6.

$$y=20 y=4$$