Solve 4-y^2-8y+25/y=16-4y/3/3

$$\frac{ 4- { y }^{ 2 } -8y+25 }{ y } = \frac{ \frac{ 16-4y }{ 3 } }{ 3 }$$

Solve for y

$$y = \frac{\sqrt{3241} - 44}{5} \approx 2.585956262$$

$$y=\frac{-\sqrt{3241}-44}{5}\approx -20.185956262$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3y, the least common multiple of y,3.

$$3\left(4-y^{2}-8y+25\right)=y\times \frac{16-4y}{3}$$

Add 4 and 25 to get 29.

$$3\left(29-y^{2}-8y\right)=y\times \frac{16-4y}{3}$$

Use the distributive property to multiply 3 by 29-y^{2}-8y.

$$87-3y^{2}-24y=y\times \frac{16-4y}{3}$$

Express y\times \frac{16-4y}{3} as a single fraction.

$$87-3y^{2}-24y=\frac{y\left(16-4y\right)}{3}$$

Use the distributive property to multiply y by 16-4y.

$$87-3y^{2}-24y=\frac{16y-4y^{2}}{3}$$

Divide each term of 16y-4y^{2} by 3 to get \frac{16}{3}y-\frac{4}{3}y^{2}.

$$87-3y^{2}-24y=\frac{16}{3}y-\frac{4}{3}y^{2}$$

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

$$87-3y^{2}-24y-\frac{16}{3}y=-\frac{4}{3}y^{2}$$

Combine -24y and -\frac{16}{3}y to get -\frac{88}{3}y.

$$87-3y^{2}-\frac{88}{3}y=-\frac{4}{3}y^{2}$$

Add \frac{4}{3}y^{2} to both sides.

$$87-3y^{2}-\frac{88}{3}y+\frac{4}{3}y^{2}=0$$

Combine -3y^{2} and \frac{4}{3}y^{2} to get -\frac{5}{3}y^{2}.

$$87-\frac{5}{3}y^{2}-\frac{88}{3}y=0$$

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.

$$-\frac{5}{3}y^{2}-\frac{88}{3}y+87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -\frac{5}{3} for a, -\frac{88}{3} for b, and 87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-\frac{88}{3}\right)±\sqrt{\left(-\frac{88}{3}\right)^{2}-4\left(-\frac{5}{3}\right)\times 87}}{2\left(-\frac{5}{3}\right)}$$

Square -\frac{88}{3} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-\frac{88}{3}\right)±\sqrt{\frac{7744}{9}-4\left(-\frac{5}{3}\right)\times 87}}{2\left(-\frac{5}{3}\right)}$$

Multiply -4 times -\frac{5}{3}.

$$y=\frac{-\left(-\frac{88}{3}\right)±\sqrt{\frac{7744}{9}+\frac{20}{3}\times 87}}{2\left(-\frac{5}{3}\right)}$$

Multiply \frac{20}{3} times 87.

$$y=\frac{-\left(-\frac{88}{3}\right)±\sqrt{\frac{7744}{9}+580}}{2\left(-\frac{5}{3}\right)}$$

Add \frac{7744}{9} to 580.

$$y=\frac{-\left(-\frac{88}{3}\right)±\sqrt{\frac{12964}{9}}}{2\left(-\frac{5}{3}\right)}$$

Take the square root of \frac{12964}{9}.

$$y=\frac{-\left(-\frac{88}{3}\right)±\frac{2\sqrt{3241}}{3}}{2\left(-\frac{5}{3}\right)}$$

The opposite of -\frac{88}{3} is \frac{88}{3}.

$$y=\frac{\frac{88}{3}±\frac{2\sqrt{3241}}{3}}{2\left(-\frac{5}{3}\right)}$$

Multiply 2 times -\frac{5}{3}.

$$y=\frac{\frac{88}{3}±\frac{2\sqrt{3241}}{3}}{-\frac{10}{3}}$$

Now solve the equation y=\frac{\frac{88}{3}±\frac{2\sqrt{3241}}{3}}{-\frac{10}{3}} when ± is plus. Add \frac{88}{3} to \frac{2\sqrt{3241}}{3}.

$$y=\frac{2\sqrt{3241}+88}{-\frac{10}{3}\times 3}$$

Divide \frac{88+2\sqrt{3241}}{3} by -\frac{10}{3} by multiplying \frac{88+2\sqrt{3241}}{3} by the reciprocal of -\frac{10}{3}.

$$y=\frac{-\sqrt{3241}-44}{5}$$

Now solve the equation y=\frac{\frac{88}{3}±\frac{2\sqrt{3241}}{3}}{-\frac{10}{3}} when ± is minus. Subtract \frac{2\sqrt{3241}}{3} from \frac{88}{3}.

$$y=\frac{88-2\sqrt{3241}}{-\frac{10}{3}\times 3}$$

Divide \frac{88-2\sqrt{3241}}{3} by -\frac{10}{3} by multiplying \frac{88-2\sqrt{3241}}{3} by the reciprocal of -\frac{10}{3}.

$$y=\frac{\sqrt{3241}-44}{5}$$

The equation is now solved.

$$y=\frac{-\sqrt{3241}-44}{5} y=\frac{\sqrt{3241}-44}{5}$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 3y, the least common multiple of y,3.

$$3\left(4-y^{2}-8y+25\right)=y\times \frac{16-4y}{3}$$

Add 4 and 25 to get 29.

$$3\left(29-y^{2}-8y\right)=y\times \frac{16-4y}{3}$$

Use the distributive property to multiply 3 by 29-y^{2}-8y.

$$87-3y^{2}-24y=y\times \frac{16-4y}{3}$$

Express y\times \frac{16-4y}{3} as a single fraction.

$$87-3y^{2}-24y=\frac{y\left(16-4y\right)}{3}$$

Use the distributive property to multiply y by 16-4y.

$$87-3y^{2}-24y=\frac{16y-4y^{2}}{3}$$

Divide each term of 16y-4y^{2} by 3 to get \frac{16}{3}y-\frac{4}{3}y^{2}.

$$87-3y^{2}-24y=\frac{16}{3}y-\frac{4}{3}y^{2}$$

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

$$87-3y^{2}-24y-\frac{16}{3}y=-\frac{4}{3}y^{2}$$

Combine -24y and -\frac{16}{3}y to get -\frac{88}{3}y.

$$87-3y^{2}-\frac{88}{3}y=-\frac{4}{3}y^{2}$$

Add \frac{4}{3}y^{2} to both sides.

$$87-3y^{2}-\frac{88}{3}y+\frac{4}{3}y^{2}=0$$

Combine -3y^{2} and \frac{4}{3}y^{2} to get -\frac{5}{3}y^{2}.

$$87-\frac{5}{3}y^{2}-\frac{88}{3}y=0$$

Subtract 87 from both sides. Anything subtracted from zero gives its negation.

$$-\frac{5}{3}y^{2}-\frac{88}{3}y=-87$$

Divide both sides of the equation by -\frac{5}{3}, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{-\frac{5}{3}y^{2}-\frac{88}{3}y}{-\frac{5}{3}}=-\frac{87}{-\frac{5}{3}}$$

Dividing by -\frac{5}{3} undoes the multiplication by -\frac{5}{3}.

$$y^{2}+\left(-\frac{\frac{88}{3}}{-\frac{5}{3}}\right)y=-\frac{87}{-\frac{5}{3}}$$

Divide -\frac{88}{3} by -\frac{5}{3} by multiplying -\frac{88}{3} by the reciprocal of -\frac{5}{3}.

$$y^{2}+\frac{88}{5}y=-\frac{87}{-\frac{5}{3}}$$

Divide -87 by -\frac{5}{3} by multiplying -87 by the reciprocal of -\frac{5}{3}.

$$y^{2}+\frac{88}{5}y=\frac{261}{5}$$

Divide \frac{88}{5}, the coefficient of the x term, by 2 to get \frac{44}{5}. Then add the square of \frac{44}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+\frac{88}{5}y+\left(\frac{44}{5}\right)^{2}=\frac{261}{5}+\left(\frac{44}{5}\right)^{2}$$

Square \frac{44}{5} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+\frac{88}{5}y+\frac{1936}{25}=\frac{261}{5}+\frac{1936}{25}$$

Add \frac{261}{5} to \frac{1936}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}+\frac{88}{5}y+\frac{1936}{25}=\frac{3241}{25}$$

Factor y^{2}+\frac{88}{5}y+\frac{1936}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y+\frac{44}{5}\right)^{2}=\frac{3241}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{44}{5}\right)^{2}}=\sqrt{\frac{3241}{25}}$$

Simplify.

$$y+\frac{44}{5}=\frac{\sqrt{3241}}{5} y+\frac{44}{5}=-\frac{\sqrt{3241}}{5}$$

Subtract \frac{44}{5} from both sides of the equation.

$$y=\frac{\sqrt{3241}-44}{5} y=\frac{-\sqrt{3241}-44}{5}$$