Solve (13/2-y)*y=-12

$$( \frac { 13 } { 2 } - y ) \cdot y = - 12$$

Solve for y

$$y=8$$

$$y = -\frac{3}{2} = -1\frac{1}{2} = -1.5$$

Steps Using the Quadratic Formula

Use the distributive property to multiply \frac{13}{2}-y by y.

$$\frac{13}{2}y-y^{2}=-12$$

Add 12 to both sides.

$$\frac{13}{2}y-y^{2}+12=0$$

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.

$$-y^{2}+\frac{13}{2}y+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, \frac{13}{2} for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\frac{13}{2}±\sqrt{\left(\frac{13}{2}\right)^{2}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Square \frac{13}{2} by squaring both the numerator and the denominator of the fraction.

$$y=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}-4\left(-1\right)\times 12}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}+4\times 12}}{2\left(-1\right)}$$

Multiply 4 times 12.

$$y=\frac{-\frac{13}{2}±\sqrt{\frac{169}{4}+48}}{2\left(-1\right)}$$

Add \frac{169}{4} to 48.

$$y=\frac{-\frac{13}{2}±\sqrt{\frac{361}{4}}}{2\left(-1\right)}$$

Take the square root of \frac{361}{4}.

$$y=\frac{-\frac{13}{2}±\frac{19}{2}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-\frac{13}{2}±\frac{19}{2}}{-2}$$

Now solve the equation y=\frac{-\frac{13}{2}±\frac{19}{2}}{-2} when ± is plus. Add -\frac{13}{2} to \frac{19}{2} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$y=\frac{3}{-2}$$

Divide 3 by -2.

$$y=-\frac{3}{2}$$

Now solve the equation y=\frac{-\frac{13}{2}±\frac{19}{2}}{-2} when ± is minus. Subtract \frac{19}{2} from -\frac{13}{2} by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$y=-\frac{16}{-2}$$

Divide -16 by -2.

$$y=8$$

The equation is now solved.

$$y=-\frac{3}{2} y=8$$

Steps for Completing the Square

Use the distributive property to multiply \frac{13}{2}-y by y.

$$\frac{13}{2}y-y^{2}=-12$$

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.

$$-y^{2}+\frac{13}{2}y=-12$$

Divide both sides by -1.

$$\frac{-y^{2}+\frac{13}{2}y}{-1}=-\frac{12}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\frac{\frac{13}{2}}{-1}y=-\frac{12}{-1}$$

Divide \frac{13}{2} by -1.

$$y^{2}-\frac{13}{2}y=-\frac{12}{-1}$$

Divide -12 by -1.

$$y^{2}-\frac{13}{2}y=12$$

Divide -\frac{13}{2}, the coefficient of the x term, by 2 to get -\frac{13}{4}. Then add the square of -\frac{13}{4} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}-\frac{13}{2}y+\left(-\frac{13}{4}\right)^{2}=12+\left(-\frac{13}{4}\right)^{2}$$

Square -\frac{13}{4} by squaring both the numerator and the denominator of the fraction.

$$y^{2}-\frac{13}{2}y+\frac{169}{16}=12+\frac{169}{16}$$

Add 12 to \frac{169}{16}.

$$y^{2}-\frac{13}{2}y+\frac{169}{16}=\frac{361}{16}$$

Factor y^{2}-\frac{13}{2}y+\frac{169}{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}.

$$\left(y-\frac{13}{4}\right)^{2}=\frac{361}{16}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y-\frac{13}{4}\right)^{2}}=\sqrt{\frac{361}{16}}$$

Simplify.

$$y-\frac{13}{4}=\frac{19}{4} y-\frac{13}{4}=-\frac{19}{4}$$

Add \frac{13}{4} to both sides of the equation.

$$y=8 y=-\frac{3}{2}$$