Solve 1.5-36y/y-36+4y=54

$$\frac{ 1.5-36y }{ y-36 } +4y = 54$$

Solve for y

$$y = \frac{\sqrt{5907} + 117}{4} \approx 48.464252523$$

$$y = \frac{117 - \sqrt{5907}}{4} \approx 10.035747477$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 36 since division by zero is not defined. Multiply both sides of the equation by y-36.

$$1.5-36y+4y\left(y-36\right)=54\left(y-36\right)$$

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

$$1.5-36y+4y^{2}-144y=54\left(y-36\right)$$

Combine -36y and -144y to get -180y.

$$1.5-180y+4y^{2}=54\left(y-36\right)$$

Use the distributive property to multiply 54 by y-36.

$$1.5-180y+4y^{2}=54y-1944$$

Subtract 54y from both sides.

$$1.5-180y+4y^{2}-54y=-1944$$

Combine -180y and -54y to get -234y.

$$1.5-234y+4y^{2}=-1944$$

Add 1944 to both sides.

$$1.5-234y+4y^{2}+1944=0$$

Add 1.5 and 1944 to get 1945.5.

$$1945.5-234y+4y^{2}=0$$

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.

$$4y^{2}-234y+1945.5=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -234 for b, and 1945.5 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-234\right)±\sqrt{\left(-234\right)^{2}-4\times 4\times 1945.5}}{2\times 4}$$

Square -234.

$$y=\frac{-\left(-234\right)±\sqrt{54756-4\times 4\times 1945.5}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-234\right)±\sqrt{54756-16\times 1945.5}}{2\times 4}$$

Multiply -16 times 1945.5.

$$y=\frac{-\left(-234\right)±\sqrt{54756-31128}}{2\times 4}$$

Add 54756 to -31128.

$$y=\frac{-\left(-234\right)±\sqrt{23628}}{2\times 4}$$

Take the square root of 23628.

$$y=\frac{-\left(-234\right)±2\sqrt{5907}}{2\times 4}$$

The opposite of -234 is 234.

$$y=\frac{234±2\sqrt{5907}}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{234±2\sqrt{5907}}{8}$$

Now solve the equation y=\frac{234±2\sqrt{5907}}{8} when ± is plus. Add 234 to 2\sqrt{5907}.

$$y=\frac{2\sqrt{5907}+234}{8}$$

Divide 234+2\sqrt{5907} by 8.

$$y=\frac{\sqrt{5907}+117}{4}$$

Now solve the equation y=\frac{234±2\sqrt{5907}}{8} when ± is minus. Subtract 2\sqrt{5907} from 234.

$$y=\frac{234-2\sqrt{5907}}{8}$$

Divide 234-2\sqrt{5907} by 8.

$$y=\frac{117-\sqrt{5907}}{4}$$

The equation is now solved.

$$y=\frac{\sqrt{5907}+117}{4} y=\frac{117-\sqrt{5907}}{4}$$

Steps for Completing the Square

Variable y cannot be equal to 36 since division by zero is not defined. Multiply both sides of the equation by y-36.

$$1.5-36y+4y\left(y-36\right)=54\left(y-36\right)$$

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

$$1.5-36y+4y^{2}-144y=54\left(y-36\right)$$

Combine -36y and -144y to get -180y.

$$1.5-180y+4y^{2}=54\left(y-36\right)$$

Use the distributive property to multiply 54 by y-36.

$$1.5-180y+4y^{2}=54y-1944$$

Subtract 54y from both sides.

$$1.5-180y+4y^{2}-54y=-1944$$

Combine -180y and -54y to get -234y.

$$1.5-234y+4y^{2}=-1944$$

Subtract 1.5 from both sides.

$$-234y+4y^{2}=-1944-1.5$$

Subtract 1.5 from -1944 to get -1945.5.

$$-234y+4y^{2}=-1945.5$$

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.

$$4y^{2}-234y=-1945.5$$

Divide both sides by 4.

$$\frac{4y^{2}-234y}{4}=-\frac{1945.5}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$y^{2}+\left(-\frac{234}{4}\right)y=-\frac{1945.5}{4}$$

Reduce the fraction \frac{-234}{4} to lowest terms by extracting and canceling out 2.

$$y^{2}-\frac{117}{2}y=-\frac{1945.5}{4}$$

Divide -1945.5 by 4.

$$y^{2}-\frac{117}{2}y=-486.375$$

Divide -\frac{117}{2}, the coefficient of the x term, by 2 to get -\frac{117}{4}. Then add the square of -\frac{117}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{117}{2}y+\left(-\frac{117}{4}\right)^{2}=-486.375+\left(-\frac{117}{4}\right)^{2}$$

Square -\frac{117}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{117}{2}y+\frac{13689}{16}=-486.375+\frac{13689}{16}$$

Add -486.375 to \frac{13689}{16} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y^{2}-\frac{117}{2}y+\frac{13689}{16}=\frac{5907}{16}$$

Factor y^{2}-\frac{117}{2}y+\frac{13689}{16}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-\frac{117}{4}\right)^{2}=\frac{5907}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{117}{4}\right)^{2}}=\sqrt{\frac{5907}{16}}$$

Simplify.

$$y-\frac{117}{4}=\frac{\sqrt{5907}}{4} y-\frac{117}{4}=-\frac{\sqrt{5907}}{4}$$

Add \frac{117}{4} to both sides of the equation.

$$y=\frac{\sqrt{5907}+117}{4} y=\frac{117-\sqrt{5907}}{4}$$