Solve 81/41-y+1.5=71/y

$$\frac { 81 } { 41 - y } + 1.5 = \frac { 71 } { y }$$

Solve for y

$$y = \frac{\sqrt{112465} + 427}{6} \approx 127.059669608$$

$$y = \frac{427 - \sqrt{112465}}{6} \approx 15.273663726$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 0,41 since division by zero is not defined. Multiply both sides of the equation by y\left(y-41\right), the least common multiple of 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 1.5=\left(y-41\right)\times 71$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-41\right)\times 1.5=\left(y-41\right)\times 71$$

Use the distributive property to multiply y by y-41.

$$-81y+\left(y^{2}-41y\right)\times 1.5=\left(y-41\right)\times 71$$

Use the distributive property to multiply y^{2}-41y by 1.5.

$$-81y+1.5y^{2}-61.5y=\left(y-41\right)\times 71$$

Combine -81y and -61.5y to get -142.5y.

$$-142.5y+1.5y^{2}=\left(y-41\right)\times 71$$

Use the distributive property to multiply y-41 by 71.

$$-142.5y+1.5y^{2}=71y-2911$$

Subtract 71y from both sides.

$$-142.5y+1.5y^{2}-71y=-2911$$

Combine -142.5y and -71y to get -213.5y.

$$-213.5y+1.5y^{2}=-2911$$

Add 2911 to both sides.

$$-213.5y+1.5y^{2}+2911=0$$

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.

$$1.5y^{2}-213.5y+2911=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5 for a, -213.5 for b, and 2911 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-213.5\right)±\sqrt{\left(-213.5\right)^{2}-4\times 1.5\times 2911}}{2\times 1.5}$$

Square -213.5 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-213.5\right)±\sqrt{45582.25-4\times 1.5\times 2911}}{2\times 1.5}$$

Multiply -4 times 1.5.

$$y=\frac{-\left(-213.5\right)±\sqrt{45582.25-6\times 2911}}{2\times 1.5}$$

Multiply -6 times 2911.

$$y=\frac{-\left(-213.5\right)±\sqrt{45582.25-17466}}{2\times 1.5}$$

Add 45582.25 to -17466.

$$y=\frac{-\left(-213.5\right)±\sqrt{28116.25}}{2\times 1.5}$$

Take the square root of 28116.25.

$$y=\frac{-\left(-213.5\right)±\frac{\sqrt{112465}}{2}}{2\times 1.5}$$

The opposite of -213.5 is 213.5.

$$y=\frac{213.5±\frac{\sqrt{112465}}{2}}{2\times 1.5}$$

Multiply 2 times 1.5.

$$y=\frac{213.5±\frac{\sqrt{112465}}{2}}{3}$$

Now solve the equation y=\frac{213.5±\frac{\sqrt{112465}}{2}}{3} when ± is plus. Add 213.5 to \frac{\sqrt{112465}}{2}.

$$y=\frac{\sqrt{112465}+427}{2\times 3}$$

Divide \frac{427+\sqrt{112465}}{2} by 3.

$$y=\frac{\sqrt{112465}+427}{6}$$

Now solve the equation y=\frac{213.5±\frac{\sqrt{112465}}{2}}{3} when ± is minus. Subtract \frac{\sqrt{112465}}{2} from 213.5.

$$y=\frac{427-\sqrt{112465}}{2\times 3}$$

Divide \frac{427-\sqrt{112465}}{2} by 3.

$$y=\frac{427-\sqrt{112465}}{6}$$

The equation is now solved.

$$y=\frac{\sqrt{112465}+427}{6} y=\frac{427-\sqrt{112465}}{6}$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 0,41 since division by zero is not defined. Multiply both sides of the equation by y\left(y-41\right), the least common multiple of 41-y,y.

$$-y\times 81+y\left(y-41\right)\times 1.5=\left(y-41\right)\times 71$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-41\right)\times 1.5=\left(y-41\right)\times 71$$

Use the distributive property to multiply y by y-41.

$$-81y+\left(y^{2}-41y\right)\times 1.5=\left(y-41\right)\times 71$$

Use the distributive property to multiply y^{2}-41y by 1.5.

$$-81y+1.5y^{2}-61.5y=\left(y-41\right)\times 71$$

Combine -81y and -61.5y to get -142.5y.

$$-142.5y+1.5y^{2}=\left(y-41\right)\times 71$$

Use the distributive property to multiply y-41 by 71.

$$-142.5y+1.5y^{2}=71y-2911$$

Subtract 71y from both sides.

$$-142.5y+1.5y^{2}-71y=-2911$$

Combine -142.5y and -71y to get -213.5y.

$$-213.5y+1.5y^{2}=-2911$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$1.5y^{2}-213.5y=-2911$$

Divide both sides of the equation by 1.5, which is the same as multiplying both sides by the reciprocal of the fraction.

$$\frac{1.5y^{2}-213.5y}{1.5}=-\frac{2911}{1.5}$$

Dividing by 1.5 undoes the multiplication by 1.5.

$$y^{2}+\left(-\frac{213.5}{1.5}\right)y=-\frac{2911}{1.5}$$

Divide -213.5 by 1.5 by multiplying -213.5 by the reciprocal of 1.5.

$$y^{2}-\frac{427}{3}y=-\frac{2911}{1.5}$$

Divide -2911 by 1.5 by multiplying -2911 by the reciprocal of 1.5.

$$y^{2}-\frac{427}{3}y=-\frac{5822}{3}$$

Divide -\frac{427}{3}, the coefficient of the x term, by 2 to get -\frac{427}{6}. Then add the square of -\frac{427}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{427}{3}y+\left(-\frac{427}{6}\right)^{2}=-\frac{5822}{3}+\left(-\frac{427}{6}\right)^{2}$$

Square -\frac{427}{6} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{427}{3}y+\frac{182329}{36}=-\frac{5822}{3}+\frac{182329}{36}$$

Add -\frac{5822}{3} to \frac{182329}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{427}{3}y+\frac{182329}{36}=\frac{112465}{36}$$

Factor y^{2}-\frac{427}{3}y+\frac{182329}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{427}{6}\right)^{2}=\frac{112465}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{427}{6}\right)^{2}}=\sqrt{\frac{112465}{36}}$$

Simplify.

$$y-\frac{427}{6}=\frac{\sqrt{112465}}{6} y-\frac{427}{6}=-\frac{\sqrt{112465}}{6}$$

Add \frac{427}{6} to both sides of the equation.

$$y=\frac{\sqrt{112465}+427}{6} y=\frac{427-\sqrt{112465}}{6}$$