Solve 3left(n+5right)^2+7=-17

$$3 { \left(n+5 \right) }^{ 2 } +7 = -17$$

Solve for n

$$n=-2\sqrt{2}i-5\approx -5-2.828427125i$$

$$n=-5+2\sqrt{2}i\approx -5+2.828427125i$$

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}.

$$3\left(n^{2}+10n+25\right)+7=-17$$

Use the distributive property to multiply 3 by n^{2}+10n+25.

$$3n^{2}+30n+75+7=-17$$

Add 75 and 7 to get 82.

$$3n^{2}+30n+82=-17$$

Add 17 to both sides.

$$3n^{2}+30n+82+17=0$$

Add 82 and 17 to get 99.

$$3n^{2}+30n+99=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 3 for a, 30 for b, and 99 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-30±\sqrt{30^{2}-4\times 3\times 99}}{2\times 3}$$

Square 30.

$$n=\frac{-30±\sqrt{900-4\times 3\times 99}}{2\times 3}$$

Multiply -4 times 3.

$$n=\frac{-30±\sqrt{900-12\times 99}}{2\times 3}$$

Multiply -12 times 99.

$$n=\frac{-30±\sqrt{900-1188}}{2\times 3}$$

Add 900 to -1188.

$$n=\frac{-30±\sqrt{-288}}{2\times 3}$$

Take the square root of -288.

$$n=\frac{-30±12\sqrt{2}i}{2\times 3}$$

Multiply 2 times 3.

$$n=\frac{-30±12\sqrt{2}i}{6}$$

Now solve the equation n=\frac{-30±12\sqrt{2}i}{6} when ± is plus. Add -30 to 12i\sqrt{2}.

$$n=\frac{-30+12\sqrt{2}i}{6}$$

Divide -30+12i\sqrt{2} by 6.

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

Now solve the equation n=\frac{-30±12\sqrt{2}i}{6} when ± is minus. Subtract 12i\sqrt{2} from -30.

$$n=\frac{-12\sqrt{2}i-30}{6}$$

Divide -30-12i\sqrt{2} by 6.

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

The equation is now solved.

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

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}.

$$3\left(n^{2}+10n+25\right)+7=-17$$

Use the distributive property to multiply 3 by n^{2}+10n+25.

$$3n^{2}+30n+75+7=-17$$

Add 75 and 7 to get 82.

$$3n^{2}+30n+82=-17$$

Subtract 82 from both sides.

$$3n^{2}+30n=-17-82$$

Subtract 82 from -17 to get -99.

$$3n^{2}+30n=-99$$

Divide both sides by 3.

$$\frac{3n^{2}+30n}{3}=-\frac{99}{3}$$

Dividing by 3 undoes the multiplication by 3.

$$n^{2}+\frac{30}{3}n=-\frac{99}{3}$$

Divide 30 by 3.

$$n^{2}+10n=-\frac{99}{3}$$

Divide -99 by 3.

$$n^{2}+10n=-33$$

Divide 10, the coefficient of the x term, by 2 to get 5. Then add the square of 5 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+10n+5^{2}=-33+5^{2}$$

Square 5.

$$n^{2}+10n+25=-33+25$$

Add -33 to 25.

$$n^{2}+10n+25=-8$$

Factor n^{2}+10n+25. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+5\right)^{2}=-8$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+5\right)^{2}}=\sqrt{-8}$$

Simplify.

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

Subtract 5 from both sides of the equation.

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