Solve x+36/x=-13

$$x + \frac { 36 } { x } = - 13$$

Solve for x

$$x=-9$$

$$x=-4$$

Steps Using Factoring

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

$$xx+36=-13x$$

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

$$x^{2}+36=-13x$$

Add 13x to both sides.

$$x^{2}+36+13x=0$$

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

$$x^{2}+13x+36=0$$

To solve the equation, factor x^{2}+13x+36 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=13 ab=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 13.

$$a=4 b=9$$

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

$$\left(x+4\right)\left(x+9\right)$$

To find equation solutions, solve x+4=0 and x+9=0.

$$x=-4 x=-9$$

Steps Using Factoring By Grouping

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

$$xx+36=-13x$$

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

$$x^{2}+36=-13x$$

Add 13x to both sides.

$$x^{2}+36+13x=0$$

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

$$x^{2}+13x+36=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+36. To find a and b, set up a system to be solved.

$$a+b=13 ab=1\times 36=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 13.

$$a=4 b=9$$

Rewrite x^{2}+13x+36 as \left(x^{2}+4x\right)+\left(9x+36\right).

$$\left(x^{2}+4x\right)+\left(9x+36\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x+4\right)+9\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+9\right)$$

To find equation solutions, solve x+4=0 and x+9=0.

$$x=-4 x=-9$$

Steps Using the Quadratic Formula

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

$$xx+36=-13x$$

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

$$x^{2}+36=-13x$$

Add 13x to both sides.

$$x^{2}+36+13x=0$$

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.

$$x^{2}+13x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 13 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-13±\sqrt{13^{2}-4\times 36}}{2}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 36}}{2}$$

Multiply -4 times 36.

$$x=\frac{-13±\sqrt{169-144}}{2}$$

Add 169 to -144.

$$x=\frac{-13±\sqrt{25}}{2}$$

Take the square root of 25.

$$x=\frac{-13±5}{2}$$

Now solve the equation x=\frac{-13±5}{2} when ± is plus. Add -13 to 5.

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

Divide -8 by 2.

$$x=-4$$

Now solve the equation x=\frac{-13±5}{2} when ± is minus. Subtract 5 from -13.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

The equation is now solved.

$$x=-4 x=-9$$

Steps for Completing the Square

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

$$xx+36=-13x$$

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

$$x^{2}+36=-13x$$

Add 13x to both sides.

$$x^{2}+36+13x=0$$

Subtract 36 from both sides. Anything subtracted from zero gives its negation.

$$x^{2}+13x=-36$$

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.

$$x^{2}+13x+\left(\frac{13}{2}\right)^{2}=-36+\left(\frac{13}{2}\right)^{2}$$

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

$$x^{2}+13x+\frac{169}{4}=-36+\frac{169}{4}$$

Add -36 to \frac{169}{4}.

$$x^{2}+13x+\frac{169}{4}=\frac{25}{4}$$

Factor x^{2}+13x+\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}.

$$\left(x+\frac{13}{2}\right)^{2}=\frac{25}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{13}{2}\right)^{2}}=\sqrt{\frac{25}{4}}$$

Simplify.

$$x+\frac{13}{2}=\frac{5}{2} x+\frac{13}{2}=-\frac{5}{2}$$

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

$$x=-4 x=-9$$