Solve 1.5*12y/12-y+4y=34

$$\frac{ 1.5 \cdot 12y }{ 12-y } +4y = 34$$

Solve for y

$$y = \frac{\sqrt{217} + 25}{2} \approx 19.865459931$$

$$y = \frac{25 - \sqrt{217}}{2} \approx 5.134540069$$

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.

$$1.5\times 12y+4y\left(-y+12\right)=34\left(-y+12\right)$$

Multiply 1.5 and 12 to get 18.

$$18y+4y\left(-y+12\right)=34\left(-y+12\right)$$

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

$$18y-4y^{2}+48y=34\left(-y+12\right)$$

Combine 18y and 48y to get 66y.

$$66y-4y^{2}=34\left(-y+12\right)$$

Use the distributive property to multiply 34 by -y+12.

$$66y-4y^{2}=-34y+408$$

Add 34y to both sides.

$$66y-4y^{2}+34y=408$$

Combine 66y and 34y to get 100y.

$$100y-4y^{2}=408$$

Subtract 408 from both sides.

$$100y-4y^{2}-408=0$$

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.

$$-4y^{2}+100y-408=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -4 for a, 100 for b, and -408 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-100±\sqrt{100^{2}-4\left(-4\right)\left(-408\right)}}{2\left(-4\right)}$$

Square 100.

$$y=\frac{-100±\sqrt{10000-4\left(-4\right)\left(-408\right)}}{2\left(-4\right)}$$

Multiply -4 times -4.

$$y=\frac{-100±\sqrt{10000+16\left(-408\right)}}{2\left(-4\right)}$$

Multiply 16 times -408.

$$y=\frac{-100±\sqrt{10000-6528}}{2\left(-4\right)}$$

Add 10000 to -6528.

$$y=\frac{-100±\sqrt{3472}}{2\left(-4\right)}$$

Take the square root of 3472.

$$y=\frac{-100±4\sqrt{217}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{-100±4\sqrt{217}}{-8}$$

Now solve the equation y=\frac{-100±4\sqrt{217}}{-8} when ± is plus. Add -100 to 4\sqrt{217}.

$$y=\frac{4\sqrt{217}-100}{-8}$$

Divide -100+4\sqrt{217} by -8.

$$y=\frac{25-\sqrt{217}}{2}$$

Now solve the equation y=\frac{-100±4\sqrt{217}}{-8} when ± is minus. Subtract 4\sqrt{217} from -100.

$$y=\frac{-4\sqrt{217}-100}{-8}$$

Divide -100-4\sqrt{217} by -8.

$$y=\frac{\sqrt{217}+25}{2}$$

The equation is now solved.

$$y=\frac{25-\sqrt{217}}{2} y=\frac{\sqrt{217}+25}{2}$$

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.

$$1.5\times 12y+4y\left(-y+12\right)=34\left(-y+12\right)$$

Multiply 1.5 and 12 to get 18.

$$18y+4y\left(-y+12\right)=34\left(-y+12\right)$$

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

$$18y-4y^{2}+48y=34\left(-y+12\right)$$

Combine 18y and 48y to get 66y.

$$66y-4y^{2}=34\left(-y+12\right)$$

Use the distributive property to multiply 34 by -y+12.

$$66y-4y^{2}=-34y+408$$

Add 34y to both sides.

$$66y-4y^{2}+34y=408$$

Combine 66y and 34y to get 100y.

$$100y-4y^{2}=408$$

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.

$$-4y^{2}+100y=408$$

Divide both sides by -4.

$$\frac{-4y^{2}+100y}{-4}=\frac{408}{-4}$$

Dividing by -4 undoes the multiplication by -4.

$$y^{2}+\frac{100}{-4}y=\frac{408}{-4}$$

Divide 100 by -4.

$$y^{2}-25y=\frac{408}{-4}$$

Divide 408 by -4.

$$y^{2}-25y=-102$$

Divide -25, the coefficient of the x term, by 2 to get -\frac{25}{2}. Then add the square of -\frac{25}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-25y+\left(-\frac{25}{2}\right)^{2}=-102+\left(-\frac{25}{2}\right)^{2}$$

Square -\frac{25}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-25y+\frac{625}{4}=-102+\frac{625}{4}$$

Add -102 to \frac{625}{4}.

$$y^{2}-25y+\frac{625}{4}=\frac{217}{4}$$

Factor y^{2}-25y+\frac{625}{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}.

$$\left(y-\frac{25}{2}\right)^{2}=\frac{217}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{25}{2}\right)^{2}}=\sqrt{\frac{217}{4}}$$

Simplify.

$$y-\frac{25}{2}=\frac{\sqrt{217}}{2} y-\frac{25}{2}=-\frac{\sqrt{217}}{2}$$

Add \frac{25}{2} to both sides of the equation.

$$y=\frac{\sqrt{217}+25}{2} y=\frac{25-\sqrt{217}}{2}$$