Solve 81/45-y+1.5=81/y

$$\frac{ 81 }{ 45-y } +1.5 = \frac{ 81 }{ y }$$

Solve for y

$$y=18$$

$$y=135$$

Steps Using the Quadratic Formula

Variable y cannot be equal to any of the values 0,45 since division by zero is not defined. Multiply both sides of the equation by y\left(y-45\right), the least common multiple of 45-y,y.

$$-y\times 81+y\left(y-45\right)\times 1.5=\left(y-45\right)\times 81$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-45\right)\times 1.5=\left(y-45\right)\times 81$$

Use the distributive property to multiply y by y-45.

$$-81y+\left(y^{2}-45y\right)\times 1.5=\left(y-45\right)\times 81$$

Use the distributive property to multiply y^{2}-45y by 1.5.

$$-81y+1.5y^{2}-67.5y=\left(y-45\right)\times 81$$

Combine -81y and -67.5y to get -148.5y.

$$-148.5y+1.5y^{2}=\left(y-45\right)\times 81$$

Use the distributive property to multiply y-45 by 81.

$$-148.5y+1.5y^{2}=81y-3645$$

Subtract 81y from both sides.

$$-148.5y+1.5y^{2}-81y=-3645$$

Combine -148.5y and -81y to get -229.5y.

$$-229.5y+1.5y^{2}=-3645$$

Add 3645 to both sides.

$$-229.5y+1.5y^{2}+3645=0$$

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.

$$1.5y^{2}-229.5y+3645=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1.5 for a, -229.5 for b, and 3645 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-229.5\right)±\sqrt{\left(-229.5\right)^{2}-4\times 1.5\times 3645}}{2\times 1.5}$$

Square -229.5 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-229.5\right)±\sqrt{52670.25-4\times 1.5\times 3645}}{2\times 1.5}$$

Multiply -4 times 1.5.

$$y=\frac{-\left(-229.5\right)±\sqrt{52670.25-6\times 3645}}{2\times 1.5}$$

Multiply -6 times 3645.

$$y=\frac{-\left(-229.5\right)±\sqrt{52670.25-21870}}{2\times 1.5}$$

Add 52670.25 to -21870.

$$y=\frac{-\left(-229.5\right)±\sqrt{30800.25}}{2\times 1.5}$$

Take the square root of 30800.25.

$$y=\frac{-\left(-229.5\right)±\frac{351}{2}}{2\times 1.5}$$

The opposite of -229.5 is 229.5.

$$y=\frac{229.5±\frac{351}{2}}{2\times 1.5}$$

Multiply 2 times 1.5.

$$y=\frac{229.5±\frac{351}{2}}{3}$$

Now solve the equation y=\frac{229.5±\frac{351}{2}}{3} when ± is plus. Add 229.5 to \frac{351}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{405}{3}$$

Divide 405 by 3.

$$y=135$$

Now solve the equation y=\frac{229.5±\frac{351}{2}}{3} when ± is minus. Subtract \frac{351}{2} from 229.5 by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{54}{3}$$

Divide 54 by 3.

$$y=18$$

The equation is now solved.

$$y=135 y=18$$

Steps for Completing the Square

Variable y cannot be equal to any of the values 0,45 since division by zero is not defined. Multiply both sides of the equation by y\left(y-45\right), the least common multiple of 45-y,y.

$$-y\times 81+y\left(y-45\right)\times 1.5=\left(y-45\right)\times 81$$

Multiply -1 and 81 to get -81.

$$-81y+y\left(y-45\right)\times 1.5=\left(y-45\right)\times 81$$

Use the distributive property to multiply y by y-45.

$$-81y+\left(y^{2}-45y\right)\times 1.5=\left(y-45\right)\times 81$$

Use the distributive property to multiply y^{2}-45y by 1.5.

$$-81y+1.5y^{2}-67.5y=\left(y-45\right)\times 81$$

Combine -81y and -67.5y to get -148.5y.

$$-148.5y+1.5y^{2}=\left(y-45\right)\times 81$$

Use the distributive property to multiply y-45 by 81.

$$-148.5y+1.5y^{2}=81y-3645$$

Subtract 81y from both sides.

$$-148.5y+1.5y^{2}-81y=-3645$$

Combine -148.5y and -81y to get -229.5y.

$$-229.5y+1.5y^{2}=-3645$$

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.

$$1.5y^{2}-229.5y=-3645$$

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

$$\frac{1.5y^{2}-229.5y}{1.5}=-\frac{3645}{1.5}$$

Dividing by 1.5 undoes the multiplication by 1.5.

$$y^{2}+\left(-\frac{229.5}{1.5}\right)y=-\frac{3645}{1.5}$$

Divide -229.5 by 1.5 by multiplying -229.5 by the reciprocal of 1.5.

$$y^{2}-153y=-\frac{3645}{1.5}$$

Divide -3645 by 1.5 by multiplying -3645 by the reciprocal of 1.5.

$$y^{2}-153y=-2430$$

Divide -153, the coefficient of the x term, by 2 to get -\frac{153}{2}. Then add the square of -\frac{153}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-153y+\left(-\frac{153}{2}\right)^{2}=-2430+\left(-\frac{153}{2}\right)^{2}$$

Square -\frac{153}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-153y+\frac{23409}{4}=-2430+\frac{23409}{4}$$

Add -2430 to \frac{23409}{4}.

$$y^{2}-153y+\frac{23409}{4}=\frac{13689}{4}$$

Factor y^{2}-153y+\frac{23409}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{153}{2}\right)^{2}=\frac{13689}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{153}{2}\right)^{2}}=\sqrt{\frac{13689}{4}}$$

Simplify.

$$y-\frac{153}{2}=\frac{117}{2} y-\frac{153}{2}=-\frac{117}{2}$$

Add \frac{153}{2} to both sides of the equation.

$$y=135 y=18$$