Solve x^2=-12x-36

$$x ^ { 2 } = - 12 x - 36$$

Solve for x

$$x=-6$$

Steps Using Factoring

Add 12x to both sides.

$$x^{2}+12x=-36$$

Add 36 to both sides.

$$x^{2}+12x+36=0$$

To solve the equation, factor x^{2}+12x+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.

$$a+b=12 ab=36$$

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 12.

$$a=6 b=6$$

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

$$\left(x+6\right)\left(x+6\right)$$

Rewrite as a binomial square.

$$\left(x+6\right)^{2}$$

To find equation solution, solve x+6=0.

$$x=-6$$

Steps Using Factoring By Grouping

Add 12x to both sides.

$$x^{2}+12x=-36$$

Add 36 to both sides.

$$x^{2}+12x+36=0$$

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.

$$a+b=12 ab=1\times 36=36$$

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 12.

$$a=6 b=6$$

Rewrite x^{2}+12x+36 as \left(x^{2}+6x\right)+\left(6x+36\right).

$$\left(x^{2}+6x\right)+\left(6x+36\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(x+6\right)+6\left(x+6\right)$$

Factor out common term x+6 by using distributive property.

$$\left(x+6\right)\left(x+6\right)$$

Rewrite as a binomial square.

$$\left(x+6\right)^{2}$$

To find equation solution, solve x+6=0.

$$x=-6$$

Steps Using the Quadratic Formula

Add 12x to both sides.

$$x^{2}+12x=-36$$

Add 36 to both sides.

$$x^{2}+12x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-12±\sqrt{12^{2}-4\times 36}}{2}$$

Square 12.

$$x=\frac{-12±\sqrt{144-4\times 36}}{2}$$

Multiply -4 times 36.

$$x=\frac{-12±\sqrt{144-144}}{2}$$

Add 144 to -144.

$$x=\frac{-12±\sqrt{0}}{2}$$

Take the square root of 0.

$$x=-\frac{12}{2}$$

Divide -12 by 2.

$$x=-6$$

Steps for Completing the Square

Add 12x to both sides.

$$x^{2}+12x=-36$$

Divide 12, the coefficient of the x term, by 2 to get 6. Then add the square of 6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+12x+6^{2}=-36+6^{2}$$

Square 6.

$$x^{2}+12x+36=-36+36$$

Add -36 to 36.

$$x^{2}+12x+36=0$$

Factor x^{2}+12x+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+6\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+6\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+6=0 x+6=0$$

Subtract 6 from both sides of the equation.

$$x=-6 x=-6$$

The equation is now solved. Solutions are the same.

$$x=-6$$