Solve a^2-2a-12

$${ a }^{ 2 } -2a-12$$

Factor

$$\left(a-\left(1-\sqrt{13}\right)\right)\left(a-\left(\sqrt{13}+1\right)\right)$$

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.

$$a^{2}-2a-12=0$$

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.

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

Square -2.

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

Multiply -4 times -12.

$$a=\frac{-\left(-2\right)±\sqrt{4+48}}{2}$$

Add 4 to 48.

$$a=\frac{-\left(-2\right)±\sqrt{52}}{2}$$

Take the square root of 52.

$$a=\frac{-\left(-2\right)±2\sqrt{13}}{2}$$

The opposite of -2 is 2.

$$a=\frac{2±2\sqrt{13}}{2}$$

Now solve the equation a=\frac{2±2\sqrt{13}}{2} when ± is plus. Add 2 to 2\sqrt{13}.

$$a=\frac{2\sqrt{13}+2}{2}$$

Divide 2+2\sqrt{13} by 2.

$$a=\sqrt{13}+1$$

Now solve the equation a=\frac{2±2\sqrt{13}}{2} when ± is minus. Subtract 2\sqrt{13} from 2.

$$a=\frac{2-2\sqrt{13}}{2}$$

Divide 2-2\sqrt{13} by 2.

$$a=1-\sqrt{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1+\sqrt{13} for x_{1} and 1-\sqrt{13} for x_{2}.

$$a^{2}-2a-12=\left(a-\left(\sqrt{13}+1\right)\right)\left(a-\left(1-\sqrt{13}\right)\right)$$

Evaluate

$$a^{2}-2a-12$$