Solve sqrt{6y+7}=5+sqrt{y-3}

$$\sqrt { 6 y + 7 } = 5 + \sqrt { y - 3 }$$

Solve for y

$$y=7$$

$$y=3$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{6y+7}\right)^{2}=\left(5+\sqrt{y-3}\right)^{2}$$

Calculate \sqrt{6y+7} to the power of 2 and get 6y+7.

$$6y+7=\left(5+\sqrt{y-3}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(5+\sqrt{y-3}\right)^{2}.

$$6y+7=25+10\sqrt{y-3}+\left(\sqrt{y-3}\right)^{2}$$

Calculate \sqrt{y-3} to the power of 2 and get y-3.

$$6y+7=25+10\sqrt{y-3}+y-3$$

Subtract 3 from 25 to get 22.

$$6y+7=22+10\sqrt{y-3}+y$$

Subtract 22+y from both sides of the equation.

$$6y+7-\left(22+y\right)=10\sqrt{y-3}$$

To find the opposite of 22+y, find the opposite of each term.

$$6y+7-22-y=10\sqrt{y-3}$$

Subtract 22 from 7 to get -15.

$$6y-15-y=10\sqrt{y-3}$$

Combine 6y and -y to get 5y.

$$5y-15=10\sqrt{y-3}$$

Square both sides of the equation.

$$\left(5y-15\right)^{2}=\left(10\sqrt{y-3}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5y-15\right)^{2}.

$$25y^{2}-150y+225=\left(10\sqrt{y-3}\right)^{2}$$

Expand \left(10\sqrt{y-3}\right)^{2}.

$$25y^{2}-150y+225=10^{2}\left(\sqrt{y-3}\right)^{2}$$

Calculate 10 to the power of 2 and get 100.

$$25y^{2}-150y+225=100\left(\sqrt{y-3}\right)^{2}$$

Calculate \sqrt{y-3} to the power of 2 and get y-3.

$$25y^{2}-150y+225=100\left(y-3\right)$$

Use the distributive property to multiply 100 by y-3.

$$25y^{2}-150y+225=100y-300$$

Subtract 100y from both sides.

$$25y^{2}-150y+225-100y=-300$$

Combine -150y and -100y to get -250y.

$$25y^{2}-250y+225=-300$$

Add 300 to both sides.

$$25y^{2}-250y+225+300=0$$

Add 225 and 300 to get 525.

$$25y^{2}-250y+525=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -250 for b, and 525 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-250\right)±\sqrt{\left(-250\right)^{2}-4\times 25\times 525}}{2\times 25}$$

Square -250.

$$y=\frac{-\left(-250\right)±\sqrt{62500-4\times 25\times 525}}{2\times 25}$$

Multiply -4 times 25.

$$y=\frac{-\left(-250\right)±\sqrt{62500-100\times 525}}{2\times 25}$$

Multiply -100 times 525.

$$y=\frac{-\left(-250\right)±\sqrt{62500-52500}}{2\times 25}$$

Add 62500 to -52500.

$$y=\frac{-\left(-250\right)±\sqrt{10000}}{2\times 25}$$

Take the square root of 10000.

$$y=\frac{-\left(-250\right)±100}{2\times 25}$$

The opposite of -250 is 250.

$$y=\frac{250±100}{2\times 25}$$

Multiply 2 times 25.

$$y=\frac{250±100}{50}$$

Now solve the equation y=\frac{250±100}{50} when ± is plus. Add 250 to 100.

$$y=\frac{350}{50}$$

Divide 350 by 50.

$$y=7$$

Now solve the equation y=\frac{250±100}{50} when ± is minus. Subtract 100 from 250.

$$y=\frac{150}{50}$$

Divide 150 by 50.

$$y=3$$

The equation is now solved.

$$y=7 y=3$$

Substitute 7 for y in the equation \sqrt{6y+7}=5+\sqrt{y-3}.

$$\sqrt{6\times 7+7}=5+\sqrt{7-3}$$

Simplify. The value y=7 satisfies the equation.

$$7=7$$

Substitute 3 for y in the equation \sqrt{6y+7}=5+\sqrt{y-3}.

$$\sqrt{6\times 3+7}=5+\sqrt{3-3}$$

Simplify. The value y=3 satisfies the equation.

$$5=5$$

List all solutions of \sqrt{6y+7}=\sqrt{y-3}+5.

$$y=7 y=3$$