Solve x^2-12x-112

$${ x }^{ 2 } -12x-112$$

Factor

$$\left(x-\left(6-2\sqrt{37}\right)\right)\left(x-\left(2\sqrt{37}+6\right)\right)$$

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.

$$x^{2}-12x-112=0$$

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.

$$x=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-112\right)}}{2}$$

Square -12.

$$x=\frac{-\left(-12\right)±\sqrt{144-4\left(-112\right)}}{2}$$

Multiply -4 times -112.

$$x=\frac{-\left(-12\right)±\sqrt{144+448}}{2}$$

Add 144 to 448.

$$x=\frac{-\left(-12\right)±\sqrt{592}}{2}$$

Take the square root of 592.

$$x=\frac{-\left(-12\right)±4\sqrt{37}}{2}$$

The opposite of -12 is 12.

$$x=\frac{12±4\sqrt{37}}{2}$$

Now solve the equation x=\frac{12±4\sqrt{37}}{2} when ± is plus. Add 12 to 4\sqrt{37}.

$$x=\frac{4\sqrt{37}+12}{2}$$

Divide 12+4\sqrt{37} by 2.

$$x=2\sqrt{37}+6$$

Now solve the equation x=\frac{12±4\sqrt{37}}{2} when ± is minus. Subtract 4\sqrt{37} from 12.

$$x=\frac{12-4\sqrt{37}}{2}$$

Divide 12-4\sqrt{37} by 2.

$$x=6-2\sqrt{37}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6+2\sqrt{37} for x_{1} and 6-2\sqrt{37} for x_{2}.

$$x^{2}-12x-112=\left(x-\left(2\sqrt{37}+6\right)\right)\left(x-\left(6-2\sqrt{37}\right)\right)$$

Evaluate

$$x^{2}-12x-112$$