Solve -11+8/y=y+36/y

$$- 11 + \frac { 8 } { y } = y + \frac { 36 } { y }$$

Solve for y

$$y=-7$$

$$y=-4$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$y\left(-11\right)+8=yy+36$$

Multiply y and y to get y^{2}.

$$y\left(-11\right)+8=y^{2}+36$$

Subtract y^{2} from both sides.

$$y\left(-11\right)+8-y^{2}=36$$

Subtract 36 from both sides.

$$y\left(-11\right)+8-y^{2}-36=0$$

Subtract 36 from 8 to get -28.

$$y\left(-11\right)-28-y^{2}=0$$

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.

$$-y^{2}-11y-28=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, -11 for b, and -28 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-\left(-11\right)±\sqrt{\left(-11\right)^{2}-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Square -11.

$$y=\frac{-\left(-11\right)±\sqrt{121-4\left(-1\right)\left(-28\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-\left(-11\right)±\sqrt{121+4\left(-28\right)}}{2\left(-1\right)}$$

Multiply 4 times -28.

$$y=\frac{-\left(-11\right)±\sqrt{121-112}}{2\left(-1\right)}$$

Add 121 to -112.

$$y=\frac{-\left(-11\right)±\sqrt{9}}{2\left(-1\right)}$$

Take the square root of 9.

$$y=\frac{-\left(-11\right)±3}{2\left(-1\right)}$$

The opposite of -11 is 11.

$$y=\frac{11±3}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{11±3}{-2}$$

Now solve the equation y=\frac{11±3}{-2} when ± is plus. Add 11 to 3.

$$y=\frac{14}{-2}$$

Divide 14 by -2.

$$y=-7$$

Now solve the equation y=\frac{11±3}{-2} when ± is minus. Subtract 3 from 11.

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

Divide 8 by -2.

$$y=-4$$

The equation is now solved.

$$y=-7 y=-4$$

Steps for Completing the Square

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$y\left(-11\right)+8=yy+36$$

Multiply y and y to get y^{2}.

$$y\left(-11\right)+8=y^{2}+36$$

Subtract y^{2} from both sides.

$$y\left(-11\right)+8-y^{2}=36$$

Subtract 8 from both sides.

$$y\left(-11\right)-y^{2}=36-8$$

Subtract 8 from 36 to get 28.

$$y\left(-11\right)-y^{2}=28$$

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.

$$-y^{2}-11y=28$$

Divide both sides by -1.

$$\frac{-y^{2}-11y}{-1}=\frac{28}{-1}$$

Dividing by -1 undoes the multiplication by -1.

$$y^{2}+\left(-\frac{11}{-1}\right)y=\frac{28}{-1}$$

Divide -11 by -1.

$$y^{2}+11y=\frac{28}{-1}$$

Divide 28 by -1.

$$y^{2}+11y=-28$$

Divide 11, the coefficient of the x term, by 2 to get \frac{11}{2}. Then add the square of \frac{11}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$y^{2}+11y+\left(\frac{11}{2}\right)^{2}=-28+\left(\frac{11}{2}\right)^{2}$$

Square \frac{11}{2} by squaring both the numerator and the denominator of the fraction.

$$y^{2}+11y+\frac{121}{4}=-28+\frac{121}{4}$$

Add -28 to \frac{121}{4}.

$$y^{2}+11y+\frac{121}{4}=\frac{9}{4}$$

Factor y^{2}+11y+\frac{121}{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}.

$$\left(y+\frac{11}{2}\right)^{2}=\frac{9}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{11}{2}\right)^{2}}=\sqrt{\frac{9}{4}}$$

Simplify.

$$y+\frac{11}{2}=\frac{3}{2} y+\frac{11}{2}=-\frac{3}{2}$$

Subtract \frac{11}{2} from both sides of the equation.

$$y=-4 y=-7$$