Solve 180=n/2(90+6n-6)

$$180 = \frac { n } { 2 } ( 90 + 6 n - 6 )$$

Solve for n

$$n=\sqrt{109}-7\approx 3.440306509$$

$$n=-\sqrt{109}-7\approx -17.440306509$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$360=n\left(90+6n-6\right)$$

Subtract 6 from 90 to get 84.

$$360=n\left(84+6n\right)$$

Use the distributive property to multiply n by 84+6n.

$$360=84n+6n^{2}$$

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

$$84n+6n^{2}=360$$

Subtract 360 from both sides.

$$84n+6n^{2}-360=0$$

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.

$$6n^{2}+84n-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 6 for a, 84 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-84±\sqrt{84^{2}-4\times 6\left(-360\right)}}{2\times 6}$$

Square 84.

$$n=\frac{-84±\sqrt{7056-4\times 6\left(-360\right)}}{2\times 6}$$

Multiply -4 times 6.

$$n=\frac{-84±\sqrt{7056-24\left(-360\right)}}{2\times 6}$$

Multiply -24 times -360.

$$n=\frac{-84±\sqrt{7056+8640}}{2\times 6}$$

Add 7056 to 8640.

$$n=\frac{-84±\sqrt{15696}}{2\times 6}$$

Take the square root of 15696.

$$n=\frac{-84±12\sqrt{109}}{2\times 6}$$

Multiply 2 times 6.

$$n=\frac{-84±12\sqrt{109}}{12}$$

Now solve the equation n=\frac{-84±12\sqrt{109}}{12} when ± is plus. Add -84 to 12\sqrt{109}.

$$n=\frac{12\sqrt{109}-84}{12}$$

Divide -84+12\sqrt{109} by 12.

$$n=\sqrt{109}-7$$

Now solve the equation n=\frac{-84±12\sqrt{109}}{12} when ± is minus. Subtract 12\sqrt{109} from -84.

$$n=\frac{-12\sqrt{109}-84}{12}$$

Divide -84-12\sqrt{109} by 12.

$$n=-\sqrt{109}-7$$

The equation is now solved.

$$n=\sqrt{109}-7 n=-\sqrt{109}-7$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$360=n\left(90+6n-6\right)$$

Subtract 6 from 90 to get 84.

$$360=n\left(84+6n\right)$$

Use the distributive property to multiply n by 84+6n.

$$360=84n+6n^{2}$$

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

$$84n+6n^{2}=360$$

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.

$$6n^{2}+84n=360$$

Divide both sides by 6.

$$\frac{6n^{2}+84n}{6}=\frac{360}{6}$$

Dividing by 6 undoes the multiplication by 6.

$$n^{2}+\frac{84}{6}n=\frac{360}{6}$$

Divide 84 by 6.

$$n^{2}+14n=\frac{360}{6}$$

Divide 360 by 6.

$$n^{2}+14n=60$$

Divide 14, the coefficient of the x term, by 2 to get 7. Then add the square of 7 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+14n+7^{2}=60+7^{2}$$

Square 7.

$$n^{2}+14n+49=60+49$$

Add 60 to 49.

$$n^{2}+14n+49=109$$

Factor n^{2}+14n+49. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+7\right)^{2}=109$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+7\right)^{2}}=\sqrt{109}$$

Simplify.

$$n+7=\sqrt{109} n+7=-\sqrt{109}$$

Subtract 7 from both sides of the equation.

$$n=\sqrt{109}-7 n=-\sqrt{109}-7$$