Solve 6n^2-12n-32

$$6 { n }^{ 2 } -12n-32$$

Factor

$$6\left(n-\left(-\frac{\sqrt{57}}{3}+1\right)\right)\left(n-\left(\frac{\sqrt{57}}{3}+1\right)\right)$$

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.

$$6n^{2}-12n-32=0$$

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.

$$n=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 6\left(-32\right)}}{2\times 6}$$

Square -12.

$$n=\frac{-\left(-12\right)±\sqrt{144-4\times 6\left(-32\right)}}{2\times 6}$$

Multiply -4 times 6.

$$n=\frac{-\left(-12\right)±\sqrt{144-24\left(-32\right)}}{2\times 6}$$

Multiply -24 times -32.

$$n=\frac{-\left(-12\right)±\sqrt{144+768}}{2\times 6}$$

Add 144 to 768.

$$n=\frac{-\left(-12\right)±\sqrt{912}}{2\times 6}$$

Take the square root of 912.

$$n=\frac{-\left(-12\right)±4\sqrt{57}}{2\times 6}$$

The opposite of -12 is 12.

$$n=\frac{12±4\sqrt{57}}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{12±4\sqrt{57}}{12}$$

Now solve the equation n=\frac{12±4\sqrt{57}}{12} when ± is plus. Add 12 to 4\sqrt{57}.

$$n=\frac{4\sqrt{57}+12}{12}$$

Divide 12+4\sqrt{57} by 12.

$$n=\frac{\sqrt{57}}{3}+1$$

Now solve the equation n=\frac{12±4\sqrt{57}}{12} when ± is minus. Subtract 4\sqrt{57} from 12.

$$n=\frac{12-4\sqrt{57}}{12}$$

Divide 12-4\sqrt{57} by 12.

$$n=-\frac{\sqrt{57}}{3}+1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1+\frac{\sqrt{57}}{3} for x_{1} and 1-\frac{\sqrt{57}}{3} for x_{2}.

$$6n^{2}-12n-32=6\left(n-\left(\frac{\sqrt{57}}{3}+1\right)\right)\left(n-\left(-\frac{\sqrt{57}}{3}+1\right)\right)$$

Evaluate

$$6n^{2}-12n-32$$