Solve 6x^2-6x-32

$$6 { x }^{ 2 } -6x-32$$

Factor

$$6\left(x-\left(-\frac{\sqrt{201}}{6}+\frac{1}{2}\right)\right)\left(x-\left(\frac{\sqrt{201}}{6}+\frac{1}{2}\right)\right)$$

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.

$$6x^{2}-6x-32=0$$

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.

$$x=\frac{-\left(-6\right)±\sqrt{\left(-6\right)^{2}-4\times 6\left(-32\right)}}{2\times 6}$$

Square -6.

$$x=\frac{-\left(-6\right)±\sqrt{36-4\times 6\left(-32\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-6\right)±\sqrt{36-24\left(-32\right)}}{2\times 6}$$

Multiply -24 times -32.

$$x=\frac{-\left(-6\right)±\sqrt{36+768}}{2\times 6}$$

Add 36 to 768.

$$x=\frac{-\left(-6\right)±\sqrt{804}}{2\times 6}$$

Take the square root of 804.

$$x=\frac{-\left(-6\right)±2\sqrt{201}}{2\times 6}$$

The opposite of -6 is 6.

$$x=\frac{6±2\sqrt{201}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{6±2\sqrt{201}}{12}$$

Now solve the equation x=\frac{6±2\sqrt{201}}{12} when ± is plus. Add 6 to 2\sqrt{201}.

$$x=\frac{2\sqrt{201}+6}{12}$$

Divide 6+2\sqrt{201} by 12.

$$x=\frac{\sqrt{201}}{6}+\frac{1}{2}$$

Now solve the equation x=\frac{6±2\sqrt{201}}{12} when ± is minus. Subtract 2\sqrt{201} from 6.

$$x=\frac{6-2\sqrt{201}}{12}$$

Divide 6-2\sqrt{201} by 12.

$$x=-\frac{\sqrt{201}}{6}+\frac{1}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2}+\frac{\sqrt{201}}{6} for x_{1} and \frac{1}{2}-\frac{\sqrt{201}}{6} for x_{2}.

$$6x^{2}-6x-32=6\left(x-\left(\frac{\sqrt{201}}{6}+\frac{1}{2}\right)\right)\left(x-\left(-\frac{\sqrt{201}}{6}+\frac{1}{2}\right)\right)$$

Evaluate

$$6x^{2}-6x-32$$