Solve 8n^2+4n-16=-n^2

$$8 n ^ { 2 } + 4 n - 16 = - n ^ { 2 }$$

Solve for n

$$n = \frac{2 \sqrt{37} - 2}{9} \approx 1.129502785$$

$$n=\frac{-2\sqrt{37}-2}{9}\approx -1.573947229$$

Steps Using the Quadratic Formula

Add n^{2} to both sides.

$$8n^{2}+4n-16+n^{2}=0$$

Combine 8n^{2} and n^{2} to get 9n^{2}.

$$9n^{2}+4n-16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 4 for b, and -16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-4±\sqrt{4^{2}-4\times 9\left(-16\right)}}{2\times 9}$$

Square 4.

$$n=\frac{-4±\sqrt{16-4\times 9\left(-16\right)}}{2\times 9}$$

Multiply -4 times 9.

$$n=\frac{-4±\sqrt{16-36\left(-16\right)}}{2\times 9}$$

Multiply -36 times -16.

$$n=\frac{-4±\sqrt{16+576}}{2\times 9}$$

Add 16 to 576.

$$n=\frac{-4±\sqrt{592}}{2\times 9}$$

Take the square root of 592.

$$n=\frac{-4±4\sqrt{37}}{2\times 9}$$

Multiply 2 times 9.

$$n=\frac{-4±4\sqrt{37}}{18}$$

Now solve the equation n=\frac{-4±4\sqrt{37}}{18} when ± is plus. Add -4 to 4\sqrt{37}.

$$n=\frac{4\sqrt{37}-4}{18}$$

Divide -4+4\sqrt{37} by 18.

$$n=\frac{2\sqrt{37}-2}{9}$$

Now solve the equation n=\frac{-4±4\sqrt{37}}{18} when ± is minus. Subtract 4\sqrt{37} from -4.

$$n=\frac{-4\sqrt{37}-4}{18}$$

Divide -4-4\sqrt{37} by 18.

$$n=\frac{-2\sqrt{37}-2}{9}$$

The equation is now solved.

$$n=\frac{2\sqrt{37}-2}{9} n=\frac{-2\sqrt{37}-2}{9}$$

Steps for Completing the Square

Add n^{2} to both sides.

$$8n^{2}+4n-16+n^{2}=0$$

Combine 8n^{2} and n^{2} to get 9n^{2}.

$$9n^{2}+4n-16=0$$

Add 16 to both sides. Anything plus zero gives itself.

$$9n^{2}+4n=16$$

Divide both sides by 9.

$$\frac{9n^{2}+4n}{9}=\frac{16}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$n^{2}+\frac{4}{9}n=\frac{16}{9}$$

Divide \frac{4}{9}, the coefficient of the x term, by 2 to get \frac{2}{9}. Then add the square of \frac{2}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{4}{9}n+\left(\frac{2}{9}\right)^{2}=\frac{16}{9}+\left(\frac{2}{9}\right)^{2}$$

Square \frac{2}{9} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{4}{9}n+\frac{4}{81}=\frac{16}{9}+\frac{4}{81}$$

Add \frac{16}{9} to \frac{4}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+\frac{4}{9}n+\frac{4}{81}=\frac{148}{81}$$

Factor n^{2}+\frac{4}{9}n+\frac{4}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{2}{9}\right)^{2}=\frac{148}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{2}{9}\right)^{2}}=\sqrt{\frac{148}{81}}$$

Simplify.

$$n+\frac{2}{9}=\frac{2\sqrt{37}}{9} n+\frac{2}{9}=-\frac{2\sqrt{37}}{9}$$

Subtract \frac{2}{9} from both sides of the equation.

$$n=\frac{2\sqrt{37}-2}{9} n=\frac{-2\sqrt{37}-2}{9}$$