Solve (n+5)^2=-2n

$$( n + 5 ) ^ { 2 } = - 2 n$$

Solve for n

$$n=\sqrt{11}-6\approx -2.68337521$$

$$n=-\sqrt{11}-6\approx -9.31662479$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(n+5\right)^{2}.

$$n^{2}+10n+25=-2n$$

Add 2n to both sides.

$$n^{2}+10n+25+2n=0$$

Combine 10n and 2n to get 12n.

$$n^{2}+12n+25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 12 for b, and 25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-12±\sqrt{12^{2}-4\times 25}}{2}$$

Square 12.

$$n=\frac{-12±\sqrt{144-4\times 25}}{2}$$

Multiply -4 times 25.

$$n=\frac{-12±\sqrt{144-100}}{2}$$

Add 144 to -100.

$$n=\frac{-12±\sqrt{44}}{2}$$

Take the square root of 44.

$$n=\frac{-12±2\sqrt{11}}{2}$$

Now solve the equation n=\frac{-12±2\sqrt{11}}{2} when ± is plus. Add -12 to 2\sqrt{11}.

$$n=\frac{2\sqrt{11}-12}{2}$$

Divide -12+2\sqrt{11} by 2.

$$n=\sqrt{11}-6$$

Now solve the equation n=\frac{-12±2\sqrt{11}}{2} when ± is minus. Subtract 2\sqrt{11} from -12.

$$n=\frac{-2\sqrt{11}-12}{2}$$

Divide -12-2\sqrt{11} by 2.

$$n=-\sqrt{11}-6$$

The equation is now solved.

$$n=\sqrt{11}-6 n=-\sqrt{11}-6$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(n+5\right)^{2}.

$$n^{2}+10n+25=-2n$$

Add 2n to both sides.

$$n^{2}+10n+25+2n=0$$

Combine 10n and 2n to get 12n.

$$n^{2}+12n+25=0$$

Subtract 25 from both sides. Anything subtracted from zero gives its negation.

$$n^{2}+12n=-25$$

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.

$$n^{2}+12n+6^{2}=-25+6^{2}$$

Square 6.

$$n^{2}+12n+36=-25+36$$

Add -25 to 36.

$$n^{2}+12n+36=11$$

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}.

$$\left(n+6\right)^{2}=11$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+6\right)^{2}}=\sqrt{11}$$

Simplify.

$$n+6=\sqrt{11} n+6=-\sqrt{11}$$

Subtract 6 from both sides of the equation.

$$n=\sqrt{11}-6 n=-\sqrt{11}-6$$