Solve n^2-12n-84=-8

$$n ^ { 2 } - 12 n - 84 = - 8$$

Solve for n

$$n=4\sqrt{7}+6\approx 16.583005244$$

$$n=6-4\sqrt{7}\approx -4.583005244$$

Steps Using the Quadratic Formula

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.

$$n^{2}-12n-84=-8$$

Add 8 to both sides of the equation.

$$n^{2}-12n-84-\left(-8\right)=-8-\left(-8\right)$$

Subtracting -8 from itself leaves 0.

$$n^{2}-12n-84-\left(-8\right)=0$$

Subtract -8 from -84.

$$n^{2}-12n-76=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -12 for b, and -76 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-\left(-12\right)±\sqrt{\left(-12\right)^{2}-4\left(-76\right)}}{2}$$

Square -12.

$$n=\frac{-\left(-12\right)±\sqrt{144-4\left(-76\right)}}{2}$$

Multiply -4 times -76.

$$n=\frac{-\left(-12\right)±\sqrt{144+304}}{2}$$

Add 144 to 304.

$$n=\frac{-\left(-12\right)±\sqrt{448}}{2}$$

Take the square root of 448.

$$n=\frac{-\left(-12\right)±8\sqrt{7}}{2}$$

The opposite of -12 is 12.

$$n=\frac{12±8\sqrt{7}}{2}$$

Now solve the equation n=\frac{12±8\sqrt{7}}{2} when ± is plus. Add 12 to 8\sqrt{7}.

$$n=\frac{8\sqrt{7}+12}{2}$$

Divide 12+8\sqrt{7} by 2.

$$n=4\sqrt{7}+6$$

Now solve the equation n=\frac{12±8\sqrt{7}}{2} when ± is minus. Subtract 8\sqrt{7} from 12.

$$n=\frac{12-8\sqrt{7}}{2}$$

Divide 12-8\sqrt{7} by 2.

$$n=6-4\sqrt{7}$$

The equation is now solved.

$$n=4\sqrt{7}+6 n=6-4\sqrt{7}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$n^{2}-12n-84=-8$$

Add 84 to both sides of the equation.

$$n^{2}-12n-84-\left(-84\right)=-8-\left(-84\right)$$

Subtracting -84 from itself leaves 0.

$$n^{2}-12n=-8-\left(-84\right)$$

Subtract -84 from -8.

$$n^{2}-12n=76$$

Divide -12, the coefficient of the x term, by 2 to get -6. Then add the square of -6 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-12n+\left(-6\right)^{2}=76+\left(-6\right)^{2}$$

Square -6.

$$n^{2}-12n+36=76+36$$

Add 76 to 36.

$$n^{2}-12n+36=112$$

Factor n^{2}-12n+36. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-6\right)^{2}=112$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-6\right)^{2}}=\sqrt{112}$$

Simplify.

$$n-6=4\sqrt{7} n-6=-4\sqrt{7}$$

Add 6 to both sides of the equation.

$$n=4\sqrt{7}+6 n=6-4\sqrt{7}$$