Solve 630=nleft(7+5n+2right)/2

$$630 = \frac{ n \left( 7+5n+2 \right) }{ 2 }$$

Solve for n

$$n = -\frac{84}{5} = -16\frac{4}{5} = -16.8$$

$$n=15$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$630\times 2=n\left(7+5n+2\right)$$

Multiply 630 and 2 to get 1260.

$$1260=n\left(7+5n+2\right)$$

Add 7 and 2 to get 9.

$$1260=n\left(9+5n\right)$$

Use the distributive property to multiply n by 9+5n.

$$1260=9n+5n^{2}$$

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

$$9n+5n^{2}=1260$$

Subtract 1260 from both sides.

$$9n+5n^{2}-1260=0$$

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.

$$5n^{2}+9n-1260=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 9 for b, and -1260 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-9±\sqrt{9^{2}-4\times 5\left(-1260\right)}}{2\times 5}$$

Square 9.

$$n=\frac{-9±\sqrt{81-4\times 5\left(-1260\right)}}{2\times 5}$$

Multiply -4 times 5.

$$n=\frac{-9±\sqrt{81-20\left(-1260\right)}}{2\times 5}$$

Multiply -20 times -1260.

$$n=\frac{-9±\sqrt{81+25200}}{2\times 5}$$

Add 81 to 25200.

$$n=\frac{-9±\sqrt{25281}}{2\times 5}$$

Take the square root of 25281.

$$n=\frac{-9±159}{2\times 5}$$

Multiply 2 times 5.

$$n=\frac{-9±159}{10}$$

Now solve the equation n=\frac{-9±159}{10} when ± is plus. Add -9 to 159.

$$n=\frac{150}{10}$$

Divide 150 by 10.

$$n=15$$

Now solve the equation n=\frac{-9±159}{10} when ± is minus. Subtract 159 from -9.

$$n=-\frac{168}{10}$$

Reduce the fraction \frac{-168}{10} to lowest terms by extracting and canceling out 2.

$$n=-\frac{84}{5}$$

The equation is now solved.

$$n=15 n=-\frac{84}{5}$$

Steps for Completing the Square

Multiply both sides by 2.

$$630\times 2=n\left(7+5n+2\right)$$

Multiply 630 and 2 to get 1260.

$$1260=n\left(7+5n+2\right)$$

Add 7 and 2 to get 9.

$$1260=n\left(9+5n\right)$$

Use the distributive property to multiply n by 9+5n.

$$1260=9n+5n^{2}$$

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

$$9n+5n^{2}=1260$$

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.

$$5n^{2}+9n=1260$$

Divide both sides by 5.

$$\frac{5n^{2}+9n}{5}=\frac{1260}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$n^{2}+\frac{9}{5}n=\frac{1260}{5}$$

Divide 1260 by 5.

$$n^{2}+\frac{9}{5}n=252$$

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.

$$n^{2}+\frac{9}{5}n+\left(\frac{9}{10}\right)^{2}=252+\left(\frac{9}{10}\right)^{2}$$

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

$$n^{2}+\frac{9}{5}n+\frac{81}{100}=252+\frac{81}{100}$$

Add 252 to \frac{81}{100}.

$$n^{2}+\frac{9}{5}n+\frac{81}{100}=\frac{25281}{100}$$

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}.

$$\left(n+\frac{9}{10}\right)^{2}=\frac{25281}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{9}{10}\right)^{2}}=\sqrt{\frac{25281}{100}}$$

Simplify.

$$n+\frac{9}{10}=\frac{159}{10} n+\frac{9}{10}=-\frac{159}{10}$$

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

$$n=15 n=-\frac{84}{5}$$