Solve 110=nleft(35+40-5nright)/2

$$110 = \frac{ n \left( 35+40-5n \right) }{ 2 }$$

Solve for n

$$n=4$$

$$n=11$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

$$110\times 2=n\left(35+40-5n\right)$$

Multiply 110 and 2 to get 220.

$$220=n\left(35+40-5n\right)$$

Add 35 and 40 to get 75.

$$220=n\left(75-5n\right)$$

Use the distributive property to multiply n by 75-5n.

$$220=75n-5n^{2}$$

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

$$75n-5n^{2}=220$$

Subtract 220 from both sides.

$$75n-5n^{2}-220=0$$

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.

$$-5n^{2}+75n-220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -5 for a, 75 for b, and -220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-75±\sqrt{75^{2}-4\left(-5\right)\left(-220\right)}}{2\left(-5\right)}$$

Square 75.

$$n=\frac{-75±\sqrt{5625-4\left(-5\right)\left(-220\right)}}{2\left(-5\right)}$$

Multiply -4 times -5.

$$n=\frac{-75±\sqrt{5625+20\left(-220\right)}}{2\left(-5\right)}$$

Multiply 20 times -220.

$$n=\frac{-75±\sqrt{5625-4400}}{2\left(-5\right)}$$

Add 5625 to -4400.

$$n=\frac{-75±\sqrt{1225}}{2\left(-5\right)}$$

Take the square root of 1225.

$$n=\frac{-75±35}{2\left(-5\right)}$$

Multiply 2 times -5.

$$n=\frac{-75±35}{-10}$$

Now solve the equation n=\frac{-75±35}{-10} when ± is plus. Add -75 to 35.

$$n=-\frac{40}{-10}$$

Divide -40 by -10.

$$n=4$$

Now solve the equation n=\frac{-75±35}{-10} when ± is minus. Subtract 35 from -75.

$$n=-\frac{110}{-10}$$

Divide -110 by -10.

$$n=11$$

The equation is now solved.

$$n=4 n=11$$

Steps for Completing the Square

Multiply both sides by 2.

$$110\times 2=n\left(35+40-5n\right)$$

Multiply 110 and 2 to get 220.

$$220=n\left(35+40-5n\right)$$

Add 35 and 40 to get 75.

$$220=n\left(75-5n\right)$$

Use the distributive property to multiply n by 75-5n.

$$220=75n-5n^{2}$$

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

$$75n-5n^{2}=220$$

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.

$$-5n^{2}+75n=220$$

Divide both sides by -5.

$$\frac{-5n^{2}+75n}{-5}=\frac{220}{-5}$$

Dividing by -5 undoes the multiplication by -5.

$$n^{2}+\frac{75}{-5}n=\frac{220}{-5}$$

Divide 75 by -5.

$$n^{2}-15n=\frac{220}{-5}$$

Divide 220 by -5.

$$n^{2}-15n=-44$$

Divide -15, the coefficient of the x term, by 2 to get -\frac{15}{2}. Then add the square of -\frac{15}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-15n+\left(-\frac{15}{2}\right)^{2}=-44+\left(-\frac{15}{2}\right)^{2}$$

Square -\frac{15}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-15n+\frac{225}{4}=-44+\frac{225}{4}$$

Add -44 to \frac{225}{4}.

$$n^{2}-15n+\frac{225}{4}=\frac{49}{4}$$

Factor n^{2}-15n+\frac{225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{15}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{15}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$n-\frac{15}{2}=\frac{7}{2} n-\frac{15}{2}=-\frac{7}{2}$$

Add \frac{15}{2} to both sides of the equation.

$$n=11 n=4$$