Solve 8+48n+16n^2

$$8 + 48 n + 16 n ^ { 2 }$$

Factor

$$16\left(n-\frac{-\sqrt{7}-3}{2}\right)\left(n-\frac{\sqrt{7}-3}{2}\right)$$

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.

$$16n^{2}+48n+8=0$$

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.

$$n=\frac{-48±\sqrt{48^{2}-4\times 16\times 8}}{2\times 16}$$

Square 48.

$$n=\frac{-48±\sqrt{2304-4\times 16\times 8}}{2\times 16}$$

Multiply -4 times 16.

$$n=\frac{-48±\sqrt{2304-64\times 8}}{2\times 16}$$

Multiply -64 times 8.

$$n=\frac{-48±\sqrt{2304-512}}{2\times 16}$$

Add 2304 to -512.

$$n=\frac{-48±\sqrt{1792}}{2\times 16}$$

Take the square root of 1792.

$$n=\frac{-48±16\sqrt{7}}{2\times 16}$$

Multiply 2 times 16.

$$n=\frac{-48±16\sqrt{7}}{32}$$

Now solve the equation n=\frac{-48±16\sqrt{7}}{32} when ± is plus. Add -48 to 16\sqrt{7}.

$$n=\frac{16\sqrt{7}-48}{32}$$

Divide -48+16\sqrt{7} by 32.

$$n=\frac{\sqrt{7}-3}{2}$$

Now solve the equation n=\frac{-48±16\sqrt{7}}{32} when ± is minus. Subtract 16\sqrt{7} from -48.

$$n=\frac{-16\sqrt{7}-48}{32}$$

Divide -48-16\sqrt{7} by 32.

$$n=\frac{-\sqrt{7}-3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3+\sqrt{7}}{2} for x_{1} and \frac{-3-\sqrt{7}}{2} for x_{2}.

$$16n^{2}+48n+8=16\left(n-\frac{\sqrt{7}-3}{2}\right)\left(n-\frac{-\sqrt{7}-3}{2}\right)$$

Evaluate

$$16n^{2}+48n+8$$