Solve 0=5n^2+9n-630

$$0 = 5 { n }^{ 2 } +9n-630$$

Solve for n

$$n = \frac{3 \sqrt{1409} - 9}{10} \approx 10.360994627$$

$$n=\frac{-3\sqrt{1409}-9}{10}\approx -12.160994627$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$5n^{2}+9n-630=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 9 for b, and -630 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-9±\sqrt{9^{2}-4\times 5\left(-630\right)}}{2\times 5}$$

Square 9.

$$n=\frac{-9±\sqrt{81-4\times 5\left(-630\right)}}{2\times 5}$$

Multiply -4 times 5.

$$n=\frac{-9±\sqrt{81-20\left(-630\right)}}{2\times 5}$$

Multiply -20 times -630.

$$n=\frac{-9±\sqrt{81+12600}}{2\times 5}$$

Add 81 to 12600.

$$n=\frac{-9±\sqrt{12681}}{2\times 5}$$

Take the square root of 12681.

$$n=\frac{-9±3\sqrt{1409}}{2\times 5}$$

Multiply 2 times 5.

$$n=\frac{-9±3\sqrt{1409}}{10}$$

Now solve the equation n=\frac{-9±3\sqrt{1409}}{10} when ± is plus. Add -9 to 3\sqrt{1409}.

$$n=\frac{3\sqrt{1409}-9}{10}$$

Now solve the equation n=\frac{-9±3\sqrt{1409}}{10} when ± is minus. Subtract 3\sqrt{1409} from -9.

$$n=\frac{-3\sqrt{1409}-9}{10}$$

The equation is now solved.

$$n=\frac{3\sqrt{1409}-9}{10} n=\frac{-3\sqrt{1409}-9}{10}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$5n^{2}+9n-630=0$$

Add 630 to both sides. Anything plus zero gives itself.

$$5n^{2}+9n=630$$

Divide both sides by 5.

$$\frac{5n^{2}+9n}{5}=\frac{630}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$n^{2}+\frac{9}{5}n=\frac{630}{5}$$

Divide 630 by 5.

$$n^{2}+\frac{9}{5}n=126$$

Divide \frac{9}{5}, the coefficient of the x term, by 2 to get \frac{9}{10}. Then add the square of \frac{9}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{9}{5}n+\left(\frac{9}{10}\right)^{2}=126+\left(\frac{9}{10}\right)^{2}$$

Square \frac{9}{10} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{9}{5}n+\frac{81}{100}=126+\frac{81}{100}$$

Add 126 to \frac{81}{100}.

$$n^{2}+\frac{9}{5}n+\frac{81}{100}=\frac{12681}{100}$$

Factor n^{2}+\frac{9}{5}n+\frac{81}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{9}{10}\right)^{2}=\frac{12681}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{9}{10}\right)^{2}}=\sqrt{\frac{12681}{100}}$$

Simplify.

$$n+\frac{9}{10}=\frac{3\sqrt{1409}}{10} n+\frac{9}{10}=-\frac{3\sqrt{1409}}{10}$$

Subtract \frac{9}{10} from both sides of the equation.

$$n=\frac{3\sqrt{1409}-9}{10} n=\frac{-3\sqrt{1409}-9}{10}$$