Solve y=sqrt{(26-3y/frac{4{3}}+2y)*26-3y/4}

$$y= \sqrt{ ( \frac{ 26-3y }{ \frac{ 4 }{ 3 } } +2y) \times \frac{ 26-3y }{ 4 } }$$

Solve for y

$$y=6$$

Solution Steps

Square both sides of the equation.

$$y^{2}=\left(\sqrt{\left(\frac{26-3y}{\frac{4}{3}}+2y\right)\times \frac{26-3y}{4}}\right)^{2}$$

Use the distributive property to multiply \frac{26-3y}{\frac{4}{3}}+2y by \frac{26-3y}{4}.

$$y^{2}=\left(\sqrt{\frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+2y\times \frac{26-3y}{4}}\right)^{2}$$

Cancel out 4, the greatest common factor in 2 and 4.

$$y^{2}=\left(\sqrt{\frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+\frac{26-3y}{2}y}\right)^{2}$$

Express \frac{26-3y}{2}y as a single fraction.

$$y^{2}=\left(\sqrt{\frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}}\right)^{2}$$

Calculate \sqrt{\frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}} to the power of 2 and get \frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}.

$$y^{2}=\frac{26-3y}{\frac{4}{3}}\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Divide each term of 26-3y by \frac{4}{3} to get \frac{26}{\frac{4}{3}}+\frac{-3y}{\frac{4}{3}}.

$$y^{2}=\left(\frac{26}{\frac{4}{3}}+\frac{-3y}{\frac{4}{3}}\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Divide 26 by \frac{4}{3} by multiplying 26 by the reciprocal of \frac{4}{3}.

$$y^{2}=\left(26\times \frac{3}{4}+\frac{-3y}{\frac{4}{3}}\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Express 26\times \frac{3}{4} as a single fraction.

$$y^{2}=\left(\frac{26\times 3}{4}+\frac{-3y}{\frac{4}{3}}\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Multiply 26 and 3 to get 78.

$$y^{2}=\left(\frac{78}{4}+\frac{-3y}{\frac{4}{3}}\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Reduce the fraction \frac{78}{4} to lowest terms by extracting and canceling out 2.

$$y^{2}=\left(\frac{39}{2}+\frac{-3y}{\frac{4}{3}}\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Divide -3y by \frac{4}{3} to get -\frac{9}{4}y.

$$y^{2}=\left(\frac{39}{2}-\frac{9}{4}y\right)\times \frac{26-3y}{4}+\frac{\left(26-3y\right)y}{2}$$

Divide each term of 26-3y by 4 to get \frac{13}{2}-\frac{3}{4}y.

$$y^{2}=\left(\frac{39}{2}-\frac{9}{4}y\right)\left(\frac{13}{2}-\frac{3}{4}y\right)+\frac{\left(26-3y\right)y}{2}$$

Apply the distributive property by multiplying each term of \frac{39}{2}-\frac{9}{4}y by each term of \frac{13}{2}-\frac{3}{4}y.

$$y^{2}=\frac{39}{2}\times \frac{13}{2}+\frac{39}{2}\left(-\frac{3}{4}\right)y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y\left(-\frac{3}{4}\right)y+\frac{\left(26-3y\right)y}{2}$$

Multiply y and y to get y^{2}.

$$y^{2}=\frac{39}{2}\times \frac{13}{2}+\frac{39}{2}\left(-\frac{3}{4}\right)y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Multiply \frac{39}{2} times \frac{13}{2} by multiplying numerator times numerator and denominator times denominator.

$$y^{2}=\frac{39\times 13}{2\times 2}+\frac{39}{2}\left(-\frac{3}{4}\right)y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Do the multiplications in the fraction \frac{39\times 13}{2\times 2}.

$$y^{2}=\frac{507}{4}+\frac{39}{2}\left(-\frac{3}{4}\right)y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Multiply \frac{39}{2} times -\frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$y^{2}=\frac{507}{4}+\frac{39\left(-3\right)}{2\times 4}y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Do the multiplications in the fraction \frac{39\left(-3\right)}{2\times 4}.

$$y^{2}=\frac{507}{4}+\frac{-117}{8}y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Fraction \frac{-117}{8} can be rewritten as -\frac{117}{8} by extracting the negative sign.

$$y^{2}=\frac{507}{4}-\frac{117}{8}y-\frac{9}{4}y\times \frac{13}{2}-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Multiply -\frac{9}{4} times \frac{13}{2} by multiplying numerator times numerator and denominator times denominator.

$$y^{2}=\frac{507}{4}-\frac{117}{8}y+\frac{-9\times 13}{4\times 2}y-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Do the multiplications in the fraction \frac{-9\times 13}{4\times 2}.

$$y^{2}=\frac{507}{4}-\frac{117}{8}y+\frac{-117}{8}y-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Fraction \frac{-117}{8} can be rewritten as -\frac{117}{8} by extracting the negative sign.

$$y^{2}=\frac{507}{4}-\frac{117}{8}y-\frac{117}{8}y-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Combine -\frac{117}{8}y and -\frac{117}{8}y to get -\frac{117}{4}y.

$$y^{2}=\frac{507}{4}-\frac{117}{4}y-\frac{9}{4}y^{2}\left(-\frac{3}{4}\right)+\frac{\left(26-3y\right)y}{2}$$

Multiply -\frac{9}{4} times -\frac{3}{4} by multiplying numerator times numerator and denominator times denominator.

$$y^{2}=\frac{507}{4}-\frac{117}{4}y+\frac{-9\left(-3\right)}{4\times 4}y^{2}+\frac{\left(26-3y\right)y}{2}$$

Do the multiplications in the fraction \frac{-9\left(-3\right)}{4\times 4}.

$$y^{2}=\frac{507}{4}-\frac{117}{4}y+\frac{27}{16}y^{2}+\frac{\left(26-3y\right)y}{2}$$

Use the distributive property to multiply 26-3y by y.

$$y^{2}=\frac{507}{4}-\frac{117}{4}y+\frac{27}{16}y^{2}+\frac{26y-3y^{2}}{2}$$

Divide each term of 26y-3y^{2} by 2 to get 13y-\frac{3}{2}y^{2}.

$$y^{2}=\frac{507}{4}-\frac{117}{4}y+\frac{27}{16}y^{2}+13y-\frac{3}{2}y^{2}$$

Combine -\frac{117}{4}y and 13y to get -\frac{65}{4}y.

$$y^{2}=\frac{507}{4}-\frac{65}{4}y+\frac{27}{16}y^{2}-\frac{3}{2}y^{2}$$

Combine \frac{27}{16}y^{2} and -\frac{3}{2}y^{2} to get \frac{3}{16}y^{2}.

$$y^{2}=\frac{507}{4}-\frac{65}{4}y+\frac{3}{16}y^{2}$$

Subtract \frac{507}{4} from both sides.

$$y^{2}-\frac{507}{4}=-\frac{65}{4}y+\frac{3}{16}y^{2}$$

Add \frac{65}{4}y to both sides.

$$y^{2}-\frac{507}{4}+\frac{65}{4}y=\frac{3}{16}y^{2}$$

Subtract \frac{3}{16}y^{2} from both sides.

$$y^{2}-\frac{507}{4}+\frac{65}{4}y-\frac{3}{16}y^{2}=0$$

Combine y^{2} and -\frac{3}{16}y^{2} to get \frac{13}{16}y^{2}.

$$\frac{13}{16}y^{2}-\frac{507}{4}+\frac{65}{4}y=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$\frac{13}{16}y^{2}+\frac{65}{4}y-\frac{507}{4}=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute \frac{13}{16} for a, \frac{65}{4} for b, and -\frac{507}{4} for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{65}{4}±\sqrt{\left(\frac{65}{4}\right)^{2}-4\times \frac{13}{16}\left(-\frac{507}{4}\right)}}{2\times \frac{13}{16}}$$

Square \frac{65}{4} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{65}{4}±\sqrt{\frac{4225}{16}-4\times \frac{13}{16}\left(-\frac{507}{4}\right)}}{2\times \frac{13}{16}}$$

Multiply -4 times \frac{13}{16}.

$$y=\frac{-\frac{65}{4}±\sqrt{\frac{4225}{16}-\frac{13}{4}\left(-\frac{507}{4}\right)}}{2\times \frac{13}{16}}$$

Multiply -\frac{13}{4} times -\frac{507}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{65}{4}±\sqrt{\frac{4225+6591}{16}}}{2\times \frac{13}{16}}$$

Add \frac{4225}{16} to \frac{6591}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{-\frac{65}{4}±\sqrt{676}}{2\times \frac{13}{16}}$$

Take the square root of 676.

$$y=\frac{-\frac{65}{4}±26}{2\times \frac{13}{16}}$$

Multiply 2 times \frac{13}{16}.

$$y=\frac{-\frac{65}{4}±26}{\frac{13}{8}}$$

Now solve the equation y=\frac{-\frac{65}{4}±26}{\frac{13}{8}} when ± is plus. Add -\frac{65}{4} to 26.

$$y=\frac{\frac{39}{4}}{\frac{13}{8}}$$

Divide \frac{39}{4} by \frac{13}{8} by multiplying \frac{39}{4} by the reciprocal of \frac{13}{8}.

$$y=6$$

Now solve the equation y=\frac{-\frac{65}{4}±26}{\frac{13}{8}} when ± is minus. Subtract 26 from -\frac{65}{4}.

$$y=-\frac{\frac{169}{4}}{\frac{13}{8}}$$

Divide -\frac{169}{4} by \frac{13}{8} by multiplying -\frac{169}{4} by the reciprocal of \frac{13}{8}.

$$y=-26$$

The equation is now solved.

$$y=6 y=-26$$

Substitute 6 for y in the equation y=\sqrt{\left(\frac{26-3y}{\frac{4}{3}}+2y\right)\times \frac{26-3y}{4}}.

$$6=\sqrt{\left(\frac{26-3\times 6}{\frac{4}{3}}+2\times 6\right)\times \frac{26-3\times 6}{4}}$$

Simplify. The value y=6 satisfies the equation.

$$6=6$$

Substitute -26 for y in the equation y=\sqrt{\left(\frac{26-3y}{\frac{4}{3}}+2y\right)\times \frac{26-3y}{4}}.

$$-26=\sqrt{\left(\frac{26-3\left(-26\right)}{\frac{4}{3}}+2\left(-26\right)\right)\times \frac{26-3\left(-26\right)}{4}}$$

Simplify. The value y=-26 does not satisfy the equation because the left and the right hand side have opposite signs.

$$-26=26$$

Equation y=\sqrt{\frac{26-3y}{4}\left(\frac{26-3y}{\frac{4}{3}}+2y\right)} has a unique solution.

$$y=6$$