Solve x^2=-8x-15

$$x ^ { 2 } = - 8 x - 15$$

Solve for x

$$x=-5$$

$$x=-3$$

Steps Using Factoring

Add 8x to both sides.

$$x^{2}+8x=-15$$

Add 15 to both sides.

$$x^{2}+8x+15=0$$

To solve the equation, factor x^{2}+8x+15 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.

$$a+b=8 ab=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

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

$$\left(x+3\right)\left(x+5\right)$$

To find equation solutions, solve x+3=0 and x+5=0.

$$x=-3 x=-5$$

Steps Using Factoring By Grouping

Add 8x to both sides.

$$x^{2}+8x=-15$$

Add 15 to both sides.

$$x^{2}+8x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 15=15$$

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 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite x^{2}+8x+15 as \left(x^{2}+3x\right)+\left(5x+15\right).

$$\left(x^{2}+3x\right)+\left(5x+15\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x+3\right)+5\left(x+3\right)$$

Factor out common term x+3 by using distributive property.

$$\left(x+3\right)\left(x+5\right)$$

To find equation solutions, solve x+3=0 and x+5=0.

$$x=-3 x=-5$$

Steps Using the Quadratic Formula

Add 8x to both sides.

$$x^{2}+8x=-15$$

Add 15 to both sides.

$$x^{2}+8x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 8 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 15}}{2}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 15}}{2}$$

Multiply -4 times 15.

$$x=\frac{-8±\sqrt{64-60}}{2}$$

Add 64 to -60.

$$x=\frac{-8±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-8±2}{2}$$

Now solve the equation x=\frac{-8±2}{2} when ± is plus. Add -8 to 2.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Now solve the equation x=\frac{-8±2}{2} when ± is minus. Subtract 2 from -8.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

The equation is now solved.

$$x=-3 x=-5$$

Steps for Completing the Square

Add 8x to both sides.

$$x^{2}+8x=-15$$

Divide 8, the coefficient of the x term, by 2 to get 4. Then add the square of 4 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+8x+4^{2}=-15+4^{2}$$

Square 4.

$$x^{2}+8x+16=-15+16$$

Add -15 to 16.

$$x^{2}+8x+16=1$$

Factor x^{2}+8x+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}.

$$\left(x+4\right)^{2}=1$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+4\right)^{2}}=\sqrt{1}$$

Simplify.

$$x+4=1 x+4=-1$$

Subtract 4 from both sides of the equation.

$$x=-3 x=-5$$