Solve sqrt{5y+1}-sqrt{3y-5}=20

$$\sqrt { 5 y + 1 } - \sqrt { 3 y - 5 } = 20$$

Solve for y

$$y=20\sqrt{1486}+797\approx 1567.973410696$$

Solution Steps

Subtract -\sqrt{3y-5} from both sides of the equation.

$$\sqrt{5y+1}=20+\sqrt{3y-5}$$

Square both sides of the equation.

$$\left(\sqrt{5y+1}\right)^{2}=\left(20+\sqrt{3y-5}\right)^{2}$$

Calculate \sqrt{5y+1} to the power of 2 and get 5y+1.

$$5y+1=\left(20+\sqrt{3y-5}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(20+\sqrt{3y-5}\right)^{2}.

$$5y+1=400+40\sqrt{3y-5}+\left(\sqrt{3y-5}\right)^{2}$$

Calculate \sqrt{3y-5} to the power of 2 and get 3y-5.

$$5y+1=400+40\sqrt{3y-5}+3y-5$$

Subtract 5 from 400 to get 395.

$$5y+1=395+40\sqrt{3y-5}+3y$$

Subtract 395+3y from both sides of the equation.

$$5y+1-\left(395+3y\right)=40\sqrt{3y-5}$$

To find the opposite of 395+3y, find the opposite of each term.

$$5y+1-395-3y=40\sqrt{3y-5}$$

Subtract 395 from 1 to get -394.

$$5y-394-3y=40\sqrt{3y-5}$$

Combine 5y and -3y to get 2y.

$$2y-394=40\sqrt{3y-5}$$

Square both sides of the equation.

$$\left(2y-394\right)^{2}=\left(40\sqrt{3y-5}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(2y-394\right)^{2}.

$$4y^{2}-1576y+155236=\left(40\sqrt{3y-5}\right)^{2}$$

Expand \left(40\sqrt{3y-5}\right)^{2}.

$$4y^{2}-1576y+155236=40^{2}\left(\sqrt{3y-5}\right)^{2}$$

Calculate 40 to the power of 2 and get 1600.

$$4y^{2}-1576y+155236=1600\left(\sqrt{3y-5}\right)^{2}$$

Calculate \sqrt{3y-5} to the power of 2 and get 3y-5.

$$4y^{2}-1576y+155236=1600\left(3y-5\right)$$

Use the distributive property to multiply 1600 by 3y-5.

$$4y^{2}-1576y+155236=4800y-8000$$

Subtract 4800y from both sides.

$$4y^{2}-1576y+155236-4800y=-8000$$

Combine -1576y and -4800y to get -6376y.

$$4y^{2}-6376y+155236=-8000$$

Add 8000 to both sides.

$$4y^{2}-6376y+155236+8000=0$$

Add 155236 and 8000 to get 163236.

$$4y^{2}-6376y+163236=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -6376 for b, and 163236 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-6376\right)±\sqrt{\left(-6376\right)^{2}-4\times 4\times 163236}}{2\times 4}$$

Square -6376.

$$y=\frac{-\left(-6376\right)±\sqrt{40653376-4\times 4\times 163236}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-6376\right)±\sqrt{40653376-16\times 163236}}{2\times 4}$$

Multiply -16 times 163236.

$$y=\frac{-\left(-6376\right)±\sqrt{40653376-2611776}}{2\times 4}$$

Add 40653376 to -2611776.

$$y=\frac{-\left(-6376\right)±\sqrt{38041600}}{2\times 4}$$

Take the square root of 38041600.

$$y=\frac{-\left(-6376\right)±160\sqrt{1486}}{2\times 4}$$

The opposite of -6376 is 6376.

$$y=\frac{6376±160\sqrt{1486}}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{6376±160\sqrt{1486}}{8}$$

Now solve the equation y=\frac{6376±160\sqrt{1486}}{8} when ± is plus. Add 6376 to 160\sqrt{1486}.

$$y=\frac{160\sqrt{1486}+6376}{8}$$

Divide 6376+160\sqrt{1486} by 8.

$$y=20\sqrt{1486}+797$$

Now solve the equation y=\frac{6376±160\sqrt{1486}}{8} when ± is minus. Subtract 160\sqrt{1486} from 6376.

$$y=\frac{6376-160\sqrt{1486}}{8}$$

Divide 6376-160\sqrt{1486} by 8.

$$y=797-20\sqrt{1486}$$

The equation is now solved.

$$y=20\sqrt{1486}+797 y=797-20\sqrt{1486}$$

Substitute 20\sqrt{1486}+797 for y in the equation \sqrt{5y+1}-\sqrt{3y-5}=20.

$$\sqrt{5\left(20\sqrt{1486}+797\right)+1}-\sqrt{3\left(20\sqrt{1486}+797\right)-5}=20$$

Simplify. The value y=20\sqrt{1486}+797 satisfies the equation.

$$20=20$$

Substitute 797-20\sqrt{1486} for y in the equation \sqrt{5y+1}-\sqrt{3y-5}=20.

$$\sqrt{5\left(797-20\sqrt{1486}\right)+1}-\sqrt{3\left(797-20\sqrt{1486}\right)-5}=20$$

Simplify. The value y=797-20\sqrt{1486} does not satisfy the equation.

$$80-2\times 1486^{\frac{1}{2}}=20$$

Substitute 20\sqrt{1486}+797 for y in the equation \sqrt{5y+1}-\sqrt{3y-5}=20.

$$\sqrt{5\left(20\sqrt{1486}+797\right)+1}-\sqrt{3\left(20\sqrt{1486}+797\right)-5}=20$$

Simplify. The value y=20\sqrt{1486}+797 satisfies the equation.

$$20=20$$

Equation \sqrt{5y+1}=\sqrt{3y-5}+20 has a unique solution.

$$y=20\sqrt{1486}+797$$