Solve y+22/y=-13

$$y + \frac { 22 } { y } = - 13$$

Solve for y

$$y=-11$$

$$y=-2$$

Steps Using Factoring

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

$$yy+22=-13y$$

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

$$y^{2}+22=-13y$$

Add 13y to both sides.

$$y^{2}+22+13y=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$y^{2}+13y+22=0$$

To solve the equation, factor y^{2}+13y+22 using formula y^{2}+\left(a+b\right)y+ab=\left(y+a\right)\left(y+b\right). To find a and b, set up a system to be solved.

$$a+b=13 ab=22$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 22.

$$1,22 2,11$$

Calculate the sum for each pair.

$$1+22=23 2+11=13$$

The solution is the pair that gives sum 13.

$$a=2 b=11$$

Rewrite factored expression \left(y+a\right)\left(y+b\right) using the obtained values.

$$\left(y+2\right)\left(y+11\right)$$

To find equation solutions, solve y+2=0 and y+11=0.

$$y=-2 y=-11$$

Steps Using Factoring By Grouping

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

$$yy+22=-13y$$

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

$$y^{2}+22=-13y$$

Add 13y to both sides.

$$y^{2}+22+13y=0$$

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$y^{2}+13y+22=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as y^{2}+ay+by+22. To find a and b, set up a system to be solved.

$$a+b=13 ab=1\times 22=22$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 22.

$$1,22 2,11$$

Calculate the sum for each pair.

$$1+22=23 2+11=13$$

The solution is the pair that gives sum 13.

$$a=2 b=11$$

Rewrite y^{2}+13y+22 as \left(y^{2}+2y\right)+\left(11y+22\right).

$$\left(y^{2}+2y\right)+\left(11y+22\right)$$

Factor out y in the first and 11 in the second group.

$$y\left(y+2\right)+11\left(y+2\right)$$

Factor out common term y+2 by using distributive property.

$$\left(y+2\right)\left(y+11\right)$$

To find equation solutions, solve y+2=0 and y+11=0.

$$y=-2 y=-11$$

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.

$$yy+22=-13y$$

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

$$y^{2}+22=-13y$$

Add 13y to both sides.

$$y^{2}+22+13y=0$$

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.

$$y^{2}+13y+22=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 13 for b, and 22 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{-13±\sqrt{13^{2}-4\times 22}}{2}$$

Square 13.

$$y=\frac{-13±\sqrt{169-4\times 22}}{2}$$

Multiply -4 times 22.

$$y=\frac{-13±\sqrt{169-88}}{2}$$

Add 169 to -88.

$$y=\frac{-13±\sqrt{81}}{2}$$

Take the square root of 81.

$$y=\frac{-13±9}{2}$$

Now solve the equation y=\frac{-13±9}{2} when ± is plus. Add -13 to 9.

$$y=-\frac{4}{2}$$

Divide -4 by 2.

$$y=-2$$

Now solve the equation y=\frac{-13±9}{2} when ± is minus. Subtract 9 from -13.

$$y=-\frac{22}{2}$$

Divide -22 by 2.

$$y=-11$$

The equation is now solved.

$$y=-2 y=-11$$

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.

$$yy+22=-13y$$

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

$$y^{2}+22=-13y$$

Add 13y to both sides.

$$y^{2}+22+13y=0$$

Subtract 22 from both sides. Anything subtracted from zero gives its negation.

$$y^{2}+13y=-22$$

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

$$y^{2}+13y+\left(\frac{13}{2}\right)^{2}=-22+\left(\frac{13}{2}\right)^{2}$$

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

$$y^{2}+13y+\frac{169}{4}=-22+\frac{169}{4}$$

Add -22 to \frac{169}{4}.

$$y^{2}+13y+\frac{169}{4}=\frac{81}{4}$$

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

$$\left(y+\frac{13}{2}\right)^{2}=\frac{81}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(y+\frac{13}{2}\right)^{2}}=\sqrt{\frac{81}{4}}$$

Simplify.

$$y+\frac{13}{2}=\frac{9}{2} y+\frac{13}{2}=-\frac{9}{2}$$

Subtract \frac{13}{2} from both sides of the equation.

$$y=-2 y=-11$$