Solve n/2left(2*20+n-1right)=200

$$\frac{ n }{ 2 } \left( 2 \cdot 20+n-1 \right) =200$$

Solve for n

$$n = \frac{\sqrt{3121} - 39}{2} \approx 8.432955447$$

$$n=\frac{-\sqrt{3121}-39}{2}\approx -47.432955447$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$n\left(2\times 20+n-1\right)=400$$

Multiply 2 and 20 to get 40.

$$n\left(40+n-1\right)=400$$

Subtract 1 from 40 to get 39.

$$n\left(39+n\right)=400$$

Use the distributive property to multiply n by 39+n.

$$39n+n^{2}=400$$

Subtract 400 from both sides.

$$39n+n^{2}-400=0$$

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.

$$n^{2}+39n-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 39 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-39±\sqrt{39^{2}-4\left(-400\right)}}{2}$$

Square 39.

$$n=\frac{-39±\sqrt{1521-4\left(-400\right)}}{2}$$

Multiply -4 times -400.

$$n=\frac{-39±\sqrt{1521+1600}}{2}$$

Add 1521 to 1600.

$$n=\frac{-39±\sqrt{3121}}{2}$$

Now solve the equation n=\frac{-39±\sqrt{3121}}{2} when ± is plus. Add -39 to \sqrt{3121}.

$$n=\frac{\sqrt{3121}-39}{2}$$

Now solve the equation n=\frac{-39±\sqrt{3121}}{2} when ± is minus. Subtract \sqrt{3121} from -39.

$$n=\frac{-\sqrt{3121}-39}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{3121}-39}{2} n=\frac{-\sqrt{3121}-39}{2}$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$n\left(2\times 20+n-1\right)=400$$

Multiply 2 and 20 to get 40.

$$n\left(40+n-1\right)=400$$

Subtract 1 from 40 to get 39.

$$n\left(39+n\right)=400$$

Use the distributive property to multiply n by 39+n.

$$39n+n^{2}=400$$

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.

$$n^{2}+39n=400$$

Divide 39, the coefficient of the x term, by 2 to get \frac{39}{2}. Then add the square of \frac{39}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+39n+\left(\frac{39}{2}\right)^{2}=400+\left(\frac{39}{2}\right)^{2}$$

Square \frac{39}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+39n+\frac{1521}{4}=400+\frac{1521}{4}$$

Add 400 to \frac{1521}{4}.

$$n^{2}+39n+\frac{1521}{4}=\frac{3121}{4}$$

Factor n^{2}+39n+\frac{1521}{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}.

$$\left(n+\frac{39}{2}\right)^{2}=\frac{3121}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{39}{2}\right)^{2}}=\sqrt{\frac{3121}{4}}$$

Simplify.

$$n+\frac{39}{2}=\frac{\sqrt{3121}}{2} n+\frac{39}{2}=-\frac{\sqrt{3121}}{2}$$

Subtract \frac{39}{2} from both sides of the equation.

$$n=\frac{\sqrt{3121}-39}{2} n=\frac{-\sqrt{3121}-39}{2}$$