Solve 3x^2-13x-12

$$3 { x }^{ 2 } -13x-12$$

Factor

$$3\left(x-\frac{13-\sqrt{313}}{6}\right)\left(x-\frac{\sqrt{313}+13}{6}\right)$$

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.

$$3x^{2}-13x-12=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\times 3\left(-12\right)}}{2\times 3}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\times 3\left(-12\right)}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-\left(-13\right)±\sqrt{169-12\left(-12\right)}}{2\times 3}$$

Multiply -12 times -12.

$$x=\frac{-\left(-13\right)±\sqrt{169+144}}{2\times 3}$$

Add 169 to 144.

$$x=\frac{-\left(-13\right)±\sqrt{313}}{2\times 3}$$

The opposite of -13 is 13.

$$x=\frac{13±\sqrt{313}}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{13±\sqrt{313}}{6}$$

Now solve the equation x=\frac{13±\sqrt{313}}{6} when ± is plus. Add 13 to \sqrt{313}.

$$x=\frac{\sqrt{313}+13}{6}$$

Now solve the equation x=\frac{13±\sqrt{313}}{6} when ± is minus. Subtract \sqrt{313} from 13.

$$x=\frac{13-\sqrt{313}}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{13+\sqrt{313}}{6} for x_{1} and \frac{13-\sqrt{313}}{6} for x_{2}.

$$3x^{2}-13x-12=3\left(x-\frac{\sqrt{313}+13}{6}\right)\left(x-\frac{13-\sqrt{313}}{6}\right)$$

Evaluate

$$3x^{2}-13x-12$$