Solve 110=n(35+40*5n)/2

$$110 = \frac { n ( 35 + 40 \cdot 5 n ) } { 2 }$$

Solve for n

$$n=\frac{\sqrt{7089}-7}{80}\approx 0.964952493$$

$$n=\frac{-\sqrt{7089}-7}{80}\approx -1.139952493$$

Steps Using the Quadratic Formula

Multiply both sides by 2.

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

Multiply 110 and 2 to get 220.

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

Multiply 40 and 5 to get 200.

$$220=n\left(35+200n\right)$$

Use the distributive property to multiply n by 35+200n.

$$220=35n+200n^{2}$$

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

$$35n+200n^{2}=220$$

Subtract 220 from both sides.

$$35n+200n^{2}-220=0$$

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.

$$200n^{2}+35n-220=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 200 for a, 35 for b, and -220 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-35±\sqrt{35^{2}-4\times 200\left(-220\right)}}{2\times 200}$$

Square 35.

$$n=\frac{-35±\sqrt{1225-4\times 200\left(-220\right)}}{2\times 200}$$

Multiply -4 times 200.

$$n=\frac{-35±\sqrt{1225-800\left(-220\right)}}{2\times 200}$$

Multiply -800 times -220.

$$n=\frac{-35±\sqrt{1225+176000}}{2\times 200}$$

Add 1225 to 176000.

$$n=\frac{-35±\sqrt{177225}}{2\times 200}$$

Take the square root of 177225.

$$n=\frac{-35±5\sqrt{7089}}{2\times 200}$$

Multiply 2 times 200.

$$n=\frac{-35±5\sqrt{7089}}{400}$$

Now solve the equation n=\frac{-35±5\sqrt{7089}}{400} when ± is plus. Add -35 to 5\sqrt{7089}.

$$n=\frac{5\sqrt{7089}-35}{400}$$

Divide -35+5\sqrt{7089} by 400.

$$n=\frac{\sqrt{7089}-7}{80}$$

Now solve the equation n=\frac{-35±5\sqrt{7089}}{400} when ± is minus. Subtract 5\sqrt{7089} from -35.

$$n=\frac{-5\sqrt{7089}-35}{400}$$

Divide -35-5\sqrt{7089} by 400.

$$n=\frac{-\sqrt{7089}-7}{80}$$

The equation is now solved.

$$n=\frac{\sqrt{7089}-7}{80} n=\frac{-\sqrt{7089}-7}{80}$$

Steps for Completing the Square

Multiply both sides by 2.

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

Multiply 110 and 2 to get 220.

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

Multiply 40 and 5 to get 200.

$$220=n\left(35+200n\right)$$

Use the distributive property to multiply n by 35+200n.

$$220=35n+200n^{2}$$

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

$$35n+200n^{2}=220$$

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.

$$200n^{2}+35n=220$$

Divide both sides by 200.

$$\frac{200n^{2}+35n}{200}=\frac{220}{200}$$

Dividing by 200 undoes the multiplication by 200.

$$n^{2}+\frac{35}{200}n=\frac{220}{200}$$

Reduce the fraction \frac{35}{200} to lowest terms by extracting and canceling out 5.

$$n^{2}+\frac{7}{40}n=\frac{220}{200}$$

Reduce the fraction \frac{220}{200} to lowest terms by extracting and canceling out 20.

$$n^{2}+\frac{7}{40}n=\frac{11}{10}$$

Divide \frac{7}{40}, the coefficient of the x term, by 2 to get \frac{7}{80}. Then add the square of \frac{7}{80} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+\frac{7}{40}n+\left(\frac{7}{80}\right)^{2}=\frac{11}{10}+\left(\frac{7}{80}\right)^{2}$$

Square \frac{7}{80} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+\frac{7}{40}n+\frac{49}{6400}=\frac{11}{10}+\frac{49}{6400}$$

Add \frac{11}{10} to \frac{49}{6400} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}+\frac{7}{40}n+\frac{49}{6400}=\frac{7089}{6400}$$

Factor n^{2}+\frac{7}{40}n+\frac{49}{6400}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n+\frac{7}{80}\right)^{2}=\frac{7089}{6400}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{7}{80}\right)^{2}}=\sqrt{\frac{7089}{6400}}$$

Simplify.

$$n+\frac{7}{80}=\frac{\sqrt{7089}}{80} n+\frac{7}{80}=-\frac{\sqrt{7089}}{80}$$

Subtract \frac{7}{80} from both sides of the equation.

$$n=\frac{\sqrt{7089}-7}{80} n=\frac{-\sqrt{7089}-7}{80}$$