Solve 1.5*12y/12-y+4y=54

$$\frac{ 1.5 \cdot 12y }{ 12-y } +4y = 54$$

Solve for y

$$y=3\sqrt{7}+15\approx 22.937253933$$

$$y=15-3\sqrt{7}\approx 7.062746067$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 12 since division by zero is not defined. Multiply both sides of the equation by -y+12.

$$1.5\times 12y+4y\left(-y+12\right)=54\left(-y+12\right)$$

Multiply 1.5 and 12 to get 18.

$$18y+4y\left(-y+12\right)=54\left(-y+12\right)$$

Use the distributive property to multiply 4y by -y+12.

$$18y-4y^{2}+48y=54\left(-y+12\right)$$

Combine 18y and 48y to get 66y.

$$66y-4y^{2}=54\left(-y+12\right)$$

Use the distributive property to multiply 54 by -y+12.

$$66y-4y^{2}=-54y+648$$

Add 54y to both sides.

$$66y-4y^{2}+54y=648$$

Combine 66y and 54y to get 120y.

$$120y-4y^{2}=648$$

Subtract 648 from both sides.

$$120y-4y^{2}-648=0$$

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.

$$-4y^{2}+120y-648=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 120 for b, and -648 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-120±\sqrt{120^{2}-4\left(-4\right)\left(-648\right)}}{2\left(-4\right)}$$

Square 120.

$$y=\frac{-120±\sqrt{14400-4\left(-4\right)\left(-648\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-120±\sqrt{14400+16\left(-648\right)}}{2\left(-4\right)}$$

Multiply 16 times -648.

$$y=\frac{-120±\sqrt{14400-10368}}{2\left(-4\right)}$$

Add 14400 to -10368.

$$y=\frac{-120±\sqrt{4032}}{2\left(-4\right)}$$

Take the square root of 4032.

$$y=\frac{-120±24\sqrt{7}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-120±24\sqrt{7}}{-8}$$

Now solve the equation y=\frac{-120±24\sqrt{7}}{-8} when ± is plus. Add -120 to 24\sqrt{7}.

$$y=\frac{24\sqrt{7}-120}{-8}$$

Divide -120+24\sqrt{7} by -8.

$$y=15-3\sqrt{7}$$

Now solve the equation y=\frac{-120±24\sqrt{7}}{-8} when ± is minus. Subtract 24\sqrt{7} from -120.

$$y=\frac{-24\sqrt{7}-120}{-8}$$

Divide -120-24\sqrt{7} by -8.

$$y=3\sqrt{7}+15$$

The equation is now solved.

$$y=15-3\sqrt{7} y=3\sqrt{7}+15$$

Steps for Completing the Square

Variable y cannot be equal to 12 since division by zero is not defined. Multiply both sides of the equation by -y+12.

$$1.5\times 12y+4y\left(-y+12\right)=54\left(-y+12\right)$$

Multiply 1.5 and 12 to get 18.

$$18y+4y\left(-y+12\right)=54\left(-y+12\right)$$

Use the distributive property to multiply 4y by -y+12.

$$18y-4y^{2}+48y=54\left(-y+12\right)$$

Combine 18y and 48y to get 66y.

$$66y-4y^{2}=54\left(-y+12\right)$$

Use the distributive property to multiply 54 by -y+12.

$$66y-4y^{2}=-54y+648$$

Add 54y to both sides.

$$66y-4y^{2}+54y=648$$

Combine 66y and 54y to get 120y.

$$120y-4y^{2}=648$$

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.

$$-4y^{2}+120y=648$$

Divide both sides by -4.

$$\frac{-4y^{2}+120y}{-4}=\frac{648}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{120}{-4}y=\frac{648}{-4}$$

Divide 120 by -4.

$$y^{2}-30y=\frac{648}{-4}$$

Divide 648 by -4.

$$y^{2}-30y=-162$$

Divide -30, the coefficient of the x term, by 2 to get -15. Then add the square of -15 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-30y+\left(-15\right)^{2}=-162+\left(-15\right)^{2}$$

Square -15.

$$y^{2}-30y+225=-162+225$$

Add -162 to 225.

$$y^{2}-30y+225=63$$

Factor y^{2}-30y+225. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(y-15\right)^{2}=63$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-15\right)^{2}}=\sqrt{63}$$

Simplify.

$$y-15=3\sqrt{7} y-15=-3\sqrt{7}$$

Add 15 to both sides of the equation.

$$y=3\sqrt{7}+15 y=15-3\sqrt{7}$$