Solve 8n+2n^2-92=0

$$8 n + 2 n ^ { 2 } - 92 = 0$$

Solve for n

$$n=5\sqrt{2}-2\approx 5.071067812$$

$$n=-5\sqrt{2}-2\approx -9.071067812$$

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.

$$2n^{2}+8n-92=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 8 for b, and -92 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-8±\sqrt{8^{2}-4\times 2\left(-92\right)}}{2\times 2}$$

Square 8.

$$n=\frac{-8±\sqrt{64-4\times 2\left(-92\right)}}{2\times 2}$$

Multiply -4 times 2.

$$n=\frac{-8±\sqrt{64-8\left(-92\right)}}{2\times 2}$$

Multiply -8 times -92.

$$n=\frac{-8±\sqrt{64+736}}{2\times 2}$$

Add 64 to 736.

$$n=\frac{-8±\sqrt{800}}{2\times 2}$$

Take the square root of 800.

$$n=\frac{-8±20\sqrt{2}}{2\times 2}$$

Multiply 2 times 2.

$$n=\frac{-8±20\sqrt{2}}{4}$$

Now solve the equation n=\frac{-8±20\sqrt{2}}{4} when ± is plus. Add -8 to 20\sqrt{2}.

$$n=\frac{20\sqrt{2}-8}{4}$$

Divide -8+20\sqrt{2} by 4.

$$n=5\sqrt{2}-2$$

Now solve the equation n=\frac{-8±20\sqrt{2}}{4} when ± is minus. Subtract 20\sqrt{2} from -8.

$$n=\frac{-20\sqrt{2}-8}{4}$$

Divide -8-20\sqrt{2} by 4.

$$n=-5\sqrt{2}-2$$

The equation is now solved.

$$n=5\sqrt{2}-2 n=-5\sqrt{2}-2$$

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.

$$2n^{2}+8n-92=0$$

Add 92 to both sides of the equation.

$$2n^{2}+8n-92-\left(-92\right)=-\left(-92\right)$$

Subtracting -92 from itself leaves 0.

$$2n^{2}+8n=-\left(-92\right)$$

Subtract -92 from 0.

$$2n^{2}+8n=92$$

Divide both sides by 2.

$$\frac{2n^{2}+8n}{2}=\frac{92}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$n^{2}+\frac{8}{2}n=\frac{92}{2}$$

Divide 8 by 2.

$$n^{2}+4n=\frac{92}{2}$$

Divide 92 by 2.

$$n^{2}+4n=46$$

Divide 4, the coefficient of the x term, by 2 to get 2. Then add the square of 2 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+4n+2^{2}=46+2^{2}$$

Square 2.

$$n^{2}+4n+4=46+4$$

Add 46 to 4.

$$n^{2}+4n+4=50$$

Factor n^{2}+4n+4. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+2\right)^{2}=50$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+2\right)^{2}}=\sqrt{50}$$

Simplify.

$$n+2=5\sqrt{2} n+2=-5\sqrt{2}$$

Subtract 2 from both sides of the equation.

$$n=5\sqrt{2}-2 n=-5\sqrt{2}-2$$