Solve 3sqrt{y}-sqrt{y+1}=2

$$3 \sqrt { y } - \sqrt { y + 1 } = 2$$

Solve for y

$$y = \frac{3 {(\sqrt{3} + 2)}}{8} \approx 1.399519053$$

Solution Steps

Subtract -\sqrt{y+1} from both sides of the equation.

$$3\sqrt{y}=2+\sqrt{y+1}$$

Square both sides of the equation.

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

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

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

Calculate 3 to the power of 2 and get 9.

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

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

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

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(2+\sqrt{y+1}\right)^{2}.

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

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

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

Add 4 and 1 to get 5.

$$9y=5+4\sqrt{y+1}+y$$

Subtract 5+y from both sides of the equation.

$$9y-\left(5+y\right)=4\sqrt{y+1}$$

To find the opposite of 5+y, find the opposite of each term.

$$9y-5-y=4\sqrt{y+1}$$

Combine 9y and -y to get 8y.

$$8y-5=4\sqrt{y+1}$$

Square both sides of the equation.

$$\left(8y-5\right)^{2}=\left(4\sqrt{y+1}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8y-5\right)^{2}.

$$64y^{2}-80y+25=\left(4\sqrt{y+1}\right)^{2}$$

Expand \left(4\sqrt{y+1}\right)^{2}.

$$64y^{2}-80y+25=4^{2}\left(\sqrt{y+1}\right)^{2}$$

Calculate 4 to the power of 2 and get 16.

$$64y^{2}-80y+25=16\left(\sqrt{y+1}\right)^{2}$$

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

$$64y^{2}-80y+25=16\left(y+1\right)$$

Use the distributive property to multiply 16 by y+1.

$$64y^{2}-80y+25=16y+16$$

Subtract 16y from both sides.

$$64y^{2}-80y+25-16y=16$$

Combine -80y and -16y to get -96y.

$$64y^{2}-96y+25=16$$

Subtract 16 from both sides.

$$64y^{2}-96y+25-16=0$$

Subtract 16 from 25 to get 9.

$$64y^{2}-96y+9=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, -96 for b, and 9 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-96\right)±\sqrt{\left(-96\right)^{2}-4\times 64\times 9}}{2\times 64}$$

Square -96.

$$y=\frac{-\left(-96\right)±\sqrt{9216-4\times 64\times 9}}{2\times 64}$$

Multiply -4 times 64.

$$y=\frac{-\left(-96\right)±\sqrt{9216-256\times 9}}{2\times 64}$$

Multiply -256 times 9.

$$y=\frac{-\left(-96\right)±\sqrt{9216-2304}}{2\times 64}$$

Add 9216 to -2304.

$$y=\frac{-\left(-96\right)±\sqrt{6912}}{2\times 64}$$

Take the square root of 6912.

$$y=\frac{-\left(-96\right)±48\sqrt{3}}{2\times 64}$$

The opposite of -96 is 96.

$$y=\frac{96±48\sqrt{3}}{2\times 64}$$

Multiply 2 times 64.

$$y=\frac{96±48\sqrt{3}}{128}$$

Now solve the equation y=\frac{96±48\sqrt{3}}{128} when ± is plus. Add 96 to 48\sqrt{3}.

$$y=\frac{48\sqrt{3}+96}{128}$$

Divide 96+48\sqrt{3} by 128.

$$y=\frac{3\sqrt{3}}{8}+\frac{3}{4}$$

Now solve the equation y=\frac{96±48\sqrt{3}}{128} when ± is minus. Subtract 48\sqrt{3} from 96.

$$y=\frac{96-48\sqrt{3}}{128}$$

Divide 96-48\sqrt{3} by 128.

$$y=-\frac{3\sqrt{3}}{8}+\frac{3}{4}$$

The equation is now solved.

$$y=\frac{3\sqrt{3}}{8}+\frac{3}{4} y=-\frac{3\sqrt{3}}{8}+\frac{3}{4}$$

Substitute \frac{3\sqrt{3}}{8}+\frac{3}{4} for y in the equation 3\sqrt{y}-\sqrt{y+1}=2.

$$3\sqrt{\frac{3\sqrt{3}}{8}+\frac{3}{4}}-\sqrt{\frac{3\sqrt{3}}{8}+\frac{3}{4}+1}=2$$

Simplify. The value y=\frac{3\sqrt{3}}{8}+\frac{3}{4} satisfies the equation.

$$2=2$$

Substitute -\frac{3\sqrt{3}}{8}+\frac{3}{4} for y in the equation 3\sqrt{y}-\sqrt{y+1}=2.

$$3\sqrt{-\frac{3\sqrt{3}}{8}+\frac{3}{4}}-\sqrt{-\frac{3\sqrt{3}}{8}+\frac{3}{4}+1}=2$$

Simplify. The value y=-\frac{3\sqrt{3}}{8}+\frac{3}{4} does not satisfy the equation because the left and the right hand side have opposite signs.

$$\frac{5}{2}-\frac{3}{2}\times 3^{\frac{1}{2}}=2$$

Substitute \frac{3\sqrt{3}}{8}+\frac{3}{4} for y in the equation 3\sqrt{y}-\sqrt{y+1}=2.

$$3\sqrt{\frac{3\sqrt{3}}{8}+\frac{3}{4}}-\sqrt{\frac{3\sqrt{3}}{8}+\frac{3}{4}+1}=2$$

Simplify. The value y=\frac{3\sqrt{3}}{8}+\frac{3}{4} satisfies the equation.

$$2=2$$

Equation 3\sqrt{y}=\sqrt{y+1}+2 has a unique solution.

$$y=\frac{3\sqrt{3}}{8}+\frac{3}{4}$$