Solve sqrt{9y+1}=3+sqrt{4y-2}

$$\sqrt{ 9y+1 } = 3+ \sqrt{ 4y-2 }$$

Solve for y

$$y = \frac{6 {(\sqrt{214} + 17)}}{25} \approx 7.590897321$$

Solution Steps

Square both sides of the equation.

$$\left(\sqrt{9y+1}\right)^{2}=\left(3+\sqrt{4y-2}\right)^{2}$$

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

$$9y+1=\left(3+\sqrt{4y-2}\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(3+\sqrt{4y-2}\right)^{2}.

$$9y+1=9+6\sqrt{4y-2}+\left(\sqrt{4y-2}\right)^{2}$$

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

$$9y+1=9+6\sqrt{4y-2}+4y-2$$

Subtract 2 from 9 to get 7.

$$9y+1=7+6\sqrt{4y-2}+4y$$

Subtract 7+4y from both sides of the equation.

$$9y+1-\left(7+4y\right)=6\sqrt{4y-2}$$

To find the opposite of 7+4y, find the opposite of each term.

$$9y+1-7-4y=6\sqrt{4y-2}$$

Subtract 7 from 1 to get -6.

$$9y-6-4y=6\sqrt{4y-2}$$

Combine 9y and -4y to get 5y.

$$5y-6=6\sqrt{4y-2}$$

Square both sides of the equation.

$$\left(5y-6\right)^{2}=\left(6\sqrt{4y-2}\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(5y-6\right)^{2}.

$$25y^{2}-60y+36=\left(6\sqrt{4y-2}\right)^{2}$$

Expand \left(6\sqrt{4y-2}\right)^{2}.

$$25y^{2}-60y+36=6^{2}\left(\sqrt{4y-2}\right)^{2}$$

Calculate 6 to the power of 2 and get 36.

$$25y^{2}-60y+36=36\left(\sqrt{4y-2}\right)^{2}$$

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

$$25y^{2}-60y+36=36\left(4y-2\right)$$

Use the distributive property to multiply 36 by 4y-2.

$$25y^{2}-60y+36=144y-72$$

Subtract 144y from both sides.

$$25y^{2}-60y+36-144y=-72$$

Combine -60y and -144y to get -204y.

$$25y^{2}-204y+36=-72$$

Add 72 to both sides.

$$25y^{2}-204y+36+72=0$$

Add 36 and 72 to get 108.

$$25y^{2}-204y+108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 25 for a, -204 for b, and 108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-204\right)±\sqrt{\left(-204\right)^{2}-4\times 25\times 108}}{2\times 25}$$

Square -204.

$$y=\frac{-\left(-204\right)±\sqrt{41616-4\times 25\times 108}}{2\times 25}$$

Multiply -4 times 25.

$$y=\frac{-\left(-204\right)±\sqrt{41616-100\times 108}}{2\times 25}$$

Multiply -100 times 108.

$$y=\frac{-\left(-204\right)±\sqrt{41616-10800}}{2\times 25}$$

Add 41616 to -10800.

$$y=\frac{-\left(-204\right)±\sqrt{30816}}{2\times 25}$$

Take the square root of 30816.

$$y=\frac{-\left(-204\right)±12\sqrt{214}}{2\times 25}$$

The opposite of -204 is 204.

$$y=\frac{204±12\sqrt{214}}{2\times 25}$$

Multiply 2 times 25.

$$y=\frac{204±12\sqrt{214}}{50}$$

Now solve the equation y=\frac{204±12\sqrt{214}}{50} when ± is plus. Add 204 to 12\sqrt{214}.

$$y=\frac{12\sqrt{214}+204}{50}$$

Divide 204+12\sqrt{214} by 50.

$$y=\frac{6\sqrt{214}+102}{25}$$

Now solve the equation y=\frac{204±12\sqrt{214}}{50} when ± is minus. Subtract 12\sqrt{214} from 204.

$$y=\frac{204-12\sqrt{214}}{50}$$

Divide 204-12\sqrt{214} by 50.

$$y=\frac{102-6\sqrt{214}}{25}$$

The equation is now solved.

$$y=\frac{6\sqrt{214}+102}{25} y=\frac{102-6\sqrt{214}}{25}$$

Substitute \frac{6\sqrt{214}+102}{25} for y in the equation \sqrt{9y+1}=3+\sqrt{4y-2}.

$$\sqrt{9\times \frac{6\sqrt{214}+102}{25}+1}=3+\sqrt{4\times \frac{6\sqrt{214}+102}{25}-2}$$

Simplify. The value y=\frac{6\sqrt{214}+102}{25} satisfies the equation.

$$\frac{27}{5}+\frac{1}{5}\times 214^{\frac{1}{2}}=\frac{27}{5}+\frac{1}{5}\times 214^{\frac{1}{2}}$$

Substitute \frac{102-6\sqrt{214}}{25} for y in the equation \sqrt{9y+1}=3+\sqrt{4y-2}.

$$\sqrt{9\times \frac{102-6\sqrt{214}}{25}+1}=3+\sqrt{4\times \frac{102-6\sqrt{214}}{25}-2}$$

Simplify. The value y=\frac{102-6\sqrt{214}}{25} does not satisfy the equation.

$$\frac{27}{5}-\frac{1}{5}\times 214^{\frac{1}{2}}=\frac{3}{5}+\frac{1}{5}\times 214^{\frac{1}{2}}$$

Substitute \frac{6\sqrt{214}+102}{25} for y in the equation \sqrt{9y+1}=3+\sqrt{4y-2}.

$$\sqrt{9\times \frac{6\sqrt{214}+102}{25}+1}=3+\sqrt{4\times \frac{6\sqrt{214}+102}{25}-2}$$

Simplify. The value y=\frac{6\sqrt{214}+102}{25} satisfies the equation.

$$\frac{27}{5}+\frac{1}{5}\times 214^{\frac{1}{2}}=\frac{27}{5}+\frac{1}{5}\times 214^{\frac{1}{2}}$$

Equation \sqrt{9y+1}=\sqrt{4y-2}+3 has a unique solution.

$$y=\frac{6\sqrt{214}+102}{25}$$