Solve sqrt{9y+1}-sqrt{y+9}=4

$$\sqrt { 9 y + 1 } - \sqrt { y + 9 } = 4$$

Solve for y

$$y=7$$

Solution Steps

Subtract -\sqrt{y+9} from both sides of the equation.

$$\sqrt{9y+1}=4+\sqrt{y+9}$$

Square both sides of the equation.

$$\left(\sqrt{9y+1}\right)^{2}=\left(4+\sqrt{y+9}\right)^{2}$$

Calculate \sqrt{9y+1} to the power of 2 and get 9y+1.

$$9y+1=\left(4+\sqrt{y+9}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(4+\sqrt{y+9}\right)^{2}.

$$9y+1=16+8\sqrt{y+9}+\left(\sqrt{y+9}\right)^{2}$$

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

$$9y+1=16+8\sqrt{y+9}+y+9$$

Add 16 and 9 to get 25.

$$9y+1=25+8\sqrt{y+9}+y$$

Subtract 25+y from both sides of the equation.

$$9y+1-\left(25+y\right)=8\sqrt{y+9}$$

To find the opposite of 25+y, find the opposite of each term.

$$9y+1-25-y=8\sqrt{y+9}$$

Subtract 25 from 1 to get -24.

$$9y-24-y=8\sqrt{y+9}$$

Combine 9y and -y to get 8y.

$$8y-24=8\sqrt{y+9}$$

Square both sides of the equation.

$$\left(8y-24\right)^{2}=\left(8\sqrt{y+9}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8y-24\right)^{2}.

$$64y^{2}-384y+576=\left(8\sqrt{y+9}\right)^{2}$$

Expand \left(8\sqrt{y+9}\right)^{2}.

$$64y^{2}-384y+576=8^{2}\left(\sqrt{y+9}\right)^{2}$$

Calculate 8 to the power of 2 and get 64.

$$64y^{2}-384y+576=64\left(\sqrt{y+9}\right)^{2}$$

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

$$64y^{2}-384y+576=64\left(y+9\right)$$

Use the distributive property to multiply 64 by y+9.

$$64y^{2}-384y+576=64y+576$$

Subtract 64y from both sides.

$$64y^{2}-384y+576-64y=576$$

Combine -384y and -64y to get -448y.

$$64y^{2}-448y+576=576$$

Subtract 576 from both sides.

$$64y^{2}-448y+576-576=0$$

Subtract 576 from 576 to get 0.

$$64y^{2}-448y=0$$

Factor out y.

$$y\left(64y-448\right)=0$$

To find equation solutions, solve y=0 and 64y-448=0.

$$y=0 y=7$$

Substitute 0 for y in the equation \sqrt{9y+1}-\sqrt{y+9}=4.

$$\sqrt{9\times 0+1}-\sqrt{0+9}=4$$

Simplify. The value y=0 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-2=4$$

Substitute 7 for y in the equation \sqrt{9y+1}-\sqrt{y+9}=4.

$$\sqrt{9\times 7+1}-\sqrt{7+9}=4$$

Simplify. The value y=7 satisfies the equation.

$$4=4$$

Equation \sqrt{9y+1}=\sqrt{y+9}+4 has a unique solution.

$$y=7$$