Solve 1.3*(-36)*y/y-36+4y=54

$$\frac { 1.3 \cdot ( - 36 ) \cdot y } { y - 36 } + 4 y = 54$$

Solve for y

$$y = \frac{9 \sqrt{139} + 153}{5} \approx 51.821687021$$

$$y = \frac{153 - 9 \sqrt{139}}{5} \approx 9.378312979$$

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.

$$1.3\left(-36\right)y+4y\left(y-36\right)=54\left(y-36\right)$$

Multiply 1.3 and -36 to get -46.8.

$$-46.8y+4y\left(y-36\right)=54\left(y-36\right)$$

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

$$-46.8y+4y^{2}-144y=54\left(y-36\right)$$

Combine -46.8y and -144y to get -190.8y.

$$-190.8y+4y^{2}=54\left(y-36\right)$$

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

$$-190.8y+4y^{2}=54y-1944$$

Subtract 54y from both sides.

$$-190.8y+4y^{2}-54y=-1944$$

Combine -190.8y and -54y to get -244.8y.

$$-244.8y+4y^{2}=-1944$$

Add 1944 to both sides.

$$-244.8y+4y^{2}+1944=0$$

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.

$$4y^{2}-244.8y+1944=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -244.8 for b, and 1944 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-244.8\right)±\sqrt{\left(-244.8\right)^{2}-4\times 4\times 1944}}{2\times 4}$$

Square -244.8 by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\left(-244.8\right)±\sqrt{59927.04-4\times 4\times 1944}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-\left(-244.8\right)±\sqrt{59927.04-16\times 1944}}{2\times 4}$$

Multiply -16 times 1944.

$$y=\frac{-\left(-244.8\right)±\sqrt{59927.04-31104}}{2\times 4}$$

Add 59927.04 to -31104.

$$y=\frac{-\left(-244.8\right)±\sqrt{28823.04}}{2\times 4}$$

Take the square root of 28823.04.

$$y=\frac{-\left(-244.8\right)±\frac{72\sqrt{139}}{5}}{2\times 4}$$

The opposite of -244.8 is 244.8.

$$y=\frac{244.8±\frac{72\sqrt{139}}{5}}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{244.8±\frac{72\sqrt{139}}{5}}{8}$$

Now solve the equation y=\frac{244.8±\frac{72\sqrt{139}}{5}}{8} when ± is plus. Add 244.8 to \frac{72\sqrt{139}}{5}.

$$y=\frac{72\sqrt{139}+1224}{5\times 8}$$

Divide \frac{1224+72\sqrt{139}}{5} by 8.

$$y=\frac{9\sqrt{139}+153}{5}$$

Now solve the equation y=\frac{244.8±\frac{72\sqrt{139}}{5}}{8} when ± is minus. Subtract \frac{72\sqrt{139}}{5} from 244.8.

$$y=\frac{1224-72\sqrt{139}}{5\times 8}$$

Divide \frac{1224-72\sqrt{139}}{5} by 8.

$$y=\frac{153-9\sqrt{139}}{5}$$

The equation is now solved.

$$y=\frac{9\sqrt{139}+153}{5} y=\frac{153-9\sqrt{139}}{5}$$

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.

$$1.3\left(-36\right)y+4y\left(y-36\right)=54\left(y-36\right)$$

Multiply 1.3 and -36 to get -46.8.

$$-46.8y+4y\left(y-36\right)=54\left(y-36\right)$$

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

$$-46.8y+4y^{2}-144y=54\left(y-36\right)$$

Combine -46.8y and -144y to get -190.8y.

$$-190.8y+4y^{2}=54\left(y-36\right)$$

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

$$-190.8y+4y^{2}=54y-1944$$

Subtract 54y from both sides.

$$-190.8y+4y^{2}-54y=-1944$$

Combine -190.8y and -54y to get -244.8y.

$$-244.8y+4y^{2}=-1944$$

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.

$$4y^{2}-244.8y=-1944$$

Divide both sides by 4.

$$\frac{4y^{2}-244.8y}{4}=-\frac{1944}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$y^{2}+\left(-\frac{244.8}{4}\right)y=-\frac{1944}{4}$$

Divide -244.8 by 4.

$$y^{2}-61.2y=-\frac{1944}{4}$$

Divide -1944 by 4.

$$y^{2}-61.2y=-486$$

Divide -61.2, the coefficient of the x term, by 2 to get -30.6. Then add the square of -30.6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-61.2y+\left(-30.6\right)^{2}=-486+\left(-30.6\right)^{2}$$

Square -30.6 by squaring both the numerator and the denominator of the fraction.

$$y^{2}-61.2y+936.36=-486+936.36$$

Add -486 to 936.36.

$$y^{2}-61.2y+936.36=450.36$$

Factor y^{2}-61.2y+936.36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-30.6\right)^{2}=450.36$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-30.6\right)^{2}}=\sqrt{450.36}$$

Simplify.

$$y-30.6=\frac{9\sqrt{139}}{5} y-30.6=-\frac{9\sqrt{139}}{5}$$

Add 30.6 to both sides of the equation.

$$y=\frac{9\sqrt{139}+153}{5} y=\frac{153-9\sqrt{139}}{5}$$