Solve x^2+20x-384

$${ x }^{ 2 } +20x-384$$

Factor

$$\left(x-12\right)\left(x+32\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-384. To find a and b, set up a system to be solved.

$$a+b=20 ab=1\left(-384\right)=-384$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -384.

$$-1,384 -2,192 -3,128 -4,96 -6,64 -8,48 -12,32 -16,24$$

Calculate the sum for each pair.

$$-1+384=383 -2+192=190 -3+128=125 -4+96=92 -6+64=58 -8+48=40 -12+32=20 -16+24=8$$

The solution is the pair that gives sum 20.

$$a=-12 b=32$$

Rewrite x^{2}+20x-384 as \left(x^{2}-12x\right)+\left(32x-384\right).

$$\left(x^{2}-12x\right)+\left(32x-384\right)$$

Factor out x in the first and 32 in the second group.

$$x\left(x-12\right)+32\left(x-12\right)$$

Factor out common term x-12 by using distributive property.

$$\left(x-12\right)\left(x+32\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$x^{2}+20x-384=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\left(-384\right)}}{2}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\left(-384\right)}}{2}$$

Multiply -4 times -384.

$$x=\frac{-20±\sqrt{400+1536}}{2}$$

Add 400 to 1536.

$$x=\frac{-20±\sqrt{1936}}{2}$$

Take the square root of 1936.

$$x=\frac{-20±44}{2}$$

Now solve the equation x=\frac{-20±44}{2} when ± is plus. Add -20 to 44.

$$x=\frac{24}{2}$$

Divide 24 by 2.

$$x=12$$

Now solve the equation x=\frac{-20±44}{2} when ± is minus. Subtract 44 from -20.

$$x=-\frac{64}{2}$$

Divide -64 by 2.

$$x=-32$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12 for x_{1} and -32 for x_{2}.

$$x^{2}+20x-384=\left(x-12\right)\left(x-\left(-32\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$x^{2}+20x-384=\left(x-12\right)\left(x+32\right)$$

Evaluate

$$\left(x-12\right)\left(x+32\right)$$