Solve sqrt{4y+12}-sqrt{y-6}=6

$$\sqrt { 4 y + 12 } - \sqrt { y - 6 } = 6$$

Solve for y

$$y=22$$

$$y=6$$

Solution Steps

Subtract -\sqrt{y-6} from both sides of the equation.

$$\sqrt{4y+12}=6+\sqrt{y-6}$$

Square both sides of the equation.

$$\left(\sqrt{4y+12}\right)^{2}=\left(6+\sqrt{y-6}\right)^{2}$$

Calculate \sqrt{4y+12} to the power of 2 and get 4y+12.

$$4y+12=\left(6+\sqrt{y-6}\right)^{2}$$

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

$$4y+12=36+12\sqrt{y-6}+\left(\sqrt{y-6}\right)^{2}$$

Calculate \sqrt{y-6} to the power of 2 and get y-6.

$$4y+12=36+12\sqrt{y-6}+y-6$$

Subtract 6 from 36 to get 30.

$$4y+12=30+12\sqrt{y-6}+y$$

Subtract 30+y from both sides of the equation.

$$4y+12-\left(30+y\right)=12\sqrt{y-6}$$

To find the opposite of 30+y, find the opposite of each term.

$$4y+12-30-y=12\sqrt{y-6}$$

Subtract 30 from 12 to get -18.

$$4y-18-y=12\sqrt{y-6}$$

Combine 4y and -y to get 3y.

$$3y-18=12\sqrt{y-6}$$

Square both sides of the equation.

$$\left(3y-18\right)^{2}=\left(12\sqrt{y-6}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(3y-18\right)^{2}.

$$9y^{2}-108y+324=\left(12\sqrt{y-6}\right)^{2}$$

Expand \left(12\sqrt{y-6}\right)^{2}.

$$9y^{2}-108y+324=12^{2}\left(\sqrt{y-6}\right)^{2}$$

Calculate 12 to the power of 2 and get 144.

$$9y^{2}-108y+324=144\left(\sqrt{y-6}\right)^{2}$$

Calculate \sqrt{y-6} to the power of 2 and get y-6.

$$9y^{2}-108y+324=144\left(y-6\right)$$

Use the distributive property to multiply 144 by y-6.

$$9y^{2}-108y+324=144y-864$$

Subtract 144y from both sides.

$$9y^{2}-108y+324-144y=-864$$

Combine -108y and -144y to get -252y.

$$9y^{2}-252y+324=-864$$

Add 864 to both sides.

$$9y^{2}-252y+324+864=0$$

Add 324 and 864 to get 1188.

$$9y^{2}-252y+1188=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -252 for b, and 1188 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-252\right)±\sqrt{\left(-252\right)^{2}-4\times 9\times 1188}}{2\times 9}$$

Square -252.

$$y=\frac{-\left(-252\right)±\sqrt{63504-4\times 9\times 1188}}{2\times 9}$$

Multiply -4 times 9.

$$y=\frac{-\left(-252\right)±\sqrt{63504-36\times 1188}}{2\times 9}$$

Multiply -36 times 1188.

$$y=\frac{-\left(-252\right)±\sqrt{63504-42768}}{2\times 9}$$

Add 63504 to -42768.

$$y=\frac{-\left(-252\right)±\sqrt{20736}}{2\times 9}$$

Take the square root of 20736.

$$y=\frac{-\left(-252\right)±144}{2\times 9}$$

The opposite of -252 is 252.

$$y=\frac{252±144}{2\times 9}$$

Multiply 2 times 9.

$$y=\frac{252±144}{18}$$

Now solve the equation y=\frac{252±144}{18} when ± is plus. Add 252 to 144.

$$y=\frac{396}{18}$$

Divide 396 by 18.

$$y=22$$

Now solve the equation y=\frac{252±144}{18} when ± is minus. Subtract 144 from 252.

$$y=\frac{108}{18}$$

Divide 108 by 18.

$$y=6$$

The equation is now solved.

$$y=22 y=6$$

Substitute 22 for y in the equation \sqrt{4y+12}-\sqrt{y-6}=6.

$$\sqrt{4\times 22+12}-\sqrt{22-6}=6$$

Simplify. The value y=22 satisfies the equation.

$$6=6$$

Substitute 6 for y in the equation \sqrt{4y+12}-\sqrt{y-6}=6.

$$\sqrt{4\times 6+12}-\sqrt{6-6}=6$$

Simplify. The value y=6 satisfies the equation.

$$6=6$$

List all solutions of \sqrt{4y+12}=\sqrt{y-6}+6.

$$y=22 y=6$$