Solve x^2+36x+308=0

$${ x }^{ 2 } +36x+308=0$$

Solve for x

$$x=-22$$

$$x=-14$$

Steps Using Factoring

To solve the equation, factor x^{2}+36x+308 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.

$$a+b=36 ab=308$$

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 308.

$$1,308 2,154 4,77 7,44 11,28 14,22$$

Calculate the sum for each pair.

$$1+308=309 2+154=156 4+77=81 7+44=51 11+28=39 14+22=36$$

The solution is the pair that gives sum 36.

$$a=14 b=22$$

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

$$\left(x+14\right)\left(x+22\right)$$

To find equation solutions, solve x+14=0 and x+22=0.

$$x=-14 x=-22$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+308. To find a and b, set up a system to be solved.

$$a+b=36 ab=1\times 308=308$$

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 308.

$$1,308 2,154 4,77 7,44 11,28 14,22$$

Calculate the sum for each pair.

$$1+308=309 2+154=156 4+77=81 7+44=51 11+28=39 14+22=36$$

The solution is the pair that gives sum 36.

$$a=14 b=22$$

Rewrite x^{2}+36x+308 as \left(x^{2}+14x\right)+\left(22x+308\right).

$$\left(x^{2}+14x\right)+\left(22x+308\right)$$

Factor out x in the first and 22 in the second group.

$$x\left(x+14\right)+22\left(x+14\right)$$

Factor out common term x+14 by using distributive property.

$$\left(x+14\right)\left(x+22\right)$$

To find equation solutions, solve x+14=0 and x+22=0.

$$x=-14 x=-22$$

Steps Using the Quadratic Formula

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.

$$x^{2}+36x+308=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 36 for b, and 308 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-36±\sqrt{36^{2}-4\times 308}}{2}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 308}}{2}$$

Multiply -4 times 308.

$$x=\frac{-36±\sqrt{1296-1232}}{2}$$

Add 1296 to -1232.

$$x=\frac{-36±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-36±8}{2}$$

Now solve the equation x=\frac{-36±8}{2} when ± is plus. Add -36 to 8.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

Now solve the equation x=\frac{-36±8}{2} when ± is minus. Subtract 8 from -36.

$$x=-\frac{44}{2}$$

Divide -44 by 2.

$$x=-22$$

The equation is now solved.

$$x=-14 x=-22$$

Steps for Completing the Square

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.

$$x^{2}+36x+308=0$$

Subtract 308 from both sides of the equation.

$$x^{2}+36x+308-308=-308$$

Subtracting 308 from itself leaves 0.

$$x^{2}+36x=-308$$

Divide 36, the coefficient of the x term, by 2 to get 18. Then add the square of 18 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+36x+18^{2}=-308+18^{2}$$

Square 18.

$$x^{2}+36x+324=-308+324$$

Add -308 to 324.

$$x^{2}+36x+324=16$$

Factor x^{2}+36x+324. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+18\right)^{2}=16$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+18\right)^{2}}=\sqrt{16}$$

Simplify.

$$x+18=4 x+18=-4$$

Subtract 18 from both sides of the equation.

$$x=-14 x=-22$$