Solve n^2-12n+3^2

$$n ^ { 2 } - 12 n + 3 ^ { 2 }$$

Factor

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

Steps Using the Quadratic Formula

Calculate 3 to the power of 2 and get 9.

$$factor(n^{2}-12n+9)$$

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.

$$n^{2}-12n+9=0$$

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.

$$n=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\times 9}}{2}$$

Square -12.

$$n=\frac{-\left(-12\right)±\sqrt{144-4\times 9}}{2}$$

Multiply -4 times 9.

$$n=\frac{-\left(-12\right)±\sqrt{144-36}}{2}$$

Add 144 to -36.

$$n=\frac{-\left(-12\right)±\sqrt{108}}{2}$$

Take the square root of 108.

$$n=\frac{-\left(-12\right)±6\sqrt{3}}{2}$$

The opposite of -12 is 12.

$$n=\frac{12±6\sqrt{3}}{2}$$

Now solve the equation n=\frac{12±6\sqrt{3}}{2} when ± is plus. Add 12 to 6\sqrt{3}.

$$n=\frac{6\sqrt{3}+12}{2}$$

Divide 12+6\sqrt{3} by 2.

$$n=3\sqrt{3}+6$$

Now solve the equation n=\frac{12±6\sqrt{3}}{2} when ± is minus. Subtract 6\sqrt{3} from 12.

$$n=\frac{12-6\sqrt{3}}{2}$$

Divide 12-6\sqrt{3} by 2.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6+3\sqrt{3} for x_{1} and 6-3\sqrt{3} for x_{2}.

$$n^{2}-12n+9=\left(n-\left(3\sqrt{3}+6\right)\right)\left(n-\left(6-3\sqrt{3}\right)\right)$$

Evaluate

$$n^{2}-12n+9$$

Solution Steps

Calculate 3 to the power of 2 and get 9.

$$n^{2}-12n+9$$