Solve n^2+3n+3n+6

$$n ^ { 2 } + 3 n + 3 n + 6$$

Factor

$$\left(n-\left(-\sqrt{3}-3\right)\right)\left(n-\left(\sqrt{3}-3\right)\right)$$

Steps Using the Quadratic Formula

Combine 3n and 3n to get 6n.

$$factor(n^{2}+6n+6)$$

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.

$$n^{2}+6n+6=0$$

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.

$$n=\frac{-6±\sqrt{6^{2}-4\times 6}}{2}$$

Square 6.

$$n=\frac{-6±\sqrt{36-4\times 6}}{2}$$

Multiply -4 times 6.

$$n=\frac{-6±\sqrt{36-24}}{2}$$

Add 36 to -24.

$$n=\frac{-6±\sqrt{12}}{2}$$

Take the square root of 12.

$$n=\frac{-6±2\sqrt{3}}{2}$$

Now solve the equation n=\frac{-6±2\sqrt{3}}{2} when ± is plus. Add -6 to 2\sqrt{3}.

$$n=\frac{2\sqrt{3}-6}{2}$$

Divide -6+2\sqrt{3} by 2.

$$n=\sqrt{3}-3$$

Now solve the equation n=\frac{-6±2\sqrt{3}}{2} when ± is minus. Subtract 2\sqrt{3} from -6.

$$n=\frac{-2\sqrt{3}-6}{2}$$

Divide -6-2\sqrt{3} by 2.

$$n=-\sqrt{3}-3$$

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

$$n^{2}+6n+6=\left(n-\left(\sqrt{3}-3\right)\right)\left(n-\left(-\sqrt{3}-3\right)\right)$$

Evaluate

$$n^{2}+6n+6$$

Solution Steps

Combine 3n and 3n to get 6n.

$$n^{2}+6n+6$$