Solve n/2left(2*400+left(n-1right)200right)=1000

$$\frac{ n }{ 2 } \left( 2 \times 400+ \left( n-1 \right) 200 \right) = 1000$$

Solve for n

$$n=-5$$

$$n=2$$

Steps Using the Quadratic Formula

Multiply both sides of the equation by 2.

$$n\left(2\times 400+\left(n-1\right)\times 200\right)=2000$$

Multiply 2 and 400 to get 800.

$$n\left(800+\left(n-1\right)\times 200\right)=2000$$

Use the distributive property to multiply n-1 by 200.

$$n\left(800+200n-200\right)=2000$$

Subtract 200 from 800 to get 600.

$$n\left(600+200n\right)=2000$$

Use the distributive property to multiply n by 600+200n.

$$600n+200n^{2}=2000$$

Subtract 2000 from both sides.

$$600n+200n^{2}-2000=0$$

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.

$$200n^{2}+600n-2000=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 200 for a, 600 for b, and -2000 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-600±\sqrt{600^{2}-4\times 200\left(-2000\right)}}{2\times 200}$$

Square 600.

$$n=\frac{-600±\sqrt{360000-4\times 200\left(-2000\right)}}{2\times 200}$$

Multiply -4 times 200.

$$n=\frac{-600±\sqrt{360000-800\left(-2000\right)}}{2\times 200}$$

Multiply -800 times -2000.

$$n=\frac{-600±\sqrt{360000+1600000}}{2\times 200}$$

Add 360000 to 1600000.

$$n=\frac{-600±\sqrt{1960000}}{2\times 200}$$

Take the square root of 1960000.

$$n=\frac{-600±1400}{2\times 200}$$

Multiply 2 times 200.

$$n=\frac{-600±1400}{400}$$

Now solve the equation n=\frac{-600±1400}{400} when ± is plus. Add -600 to 1400.

$$n=\frac{800}{400}$$

Divide 800 by 400.

$$n=2$$

Now solve the equation n=\frac{-600±1400}{400} when ± is minus. Subtract 1400 from -600.

$$n=-\frac{2000}{400}$$

Divide -2000 by 400.

$$n=-5$$

The equation is now solved.

$$n=2 n=-5$$

Steps for Completing the Square

Multiply both sides of the equation by 2.

$$n\left(2\times 400+\left(n-1\right)\times 200\right)=2000$$

Multiply 2 and 400 to get 800.

$$n\left(800+\left(n-1\right)\times 200\right)=2000$$

Use the distributive property to multiply n-1 by 200.

$$n\left(800+200n-200\right)=2000$$

Subtract 200 from 800 to get 600.

$$n\left(600+200n\right)=2000$$

Use the distributive property to multiply n by 600+200n.

$$600n+200n^{2}=2000$$

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.

$$200n^{2}+600n=2000$$

Divide both sides by 200.

$$\frac{200n^{2}+600n}{200}=\frac{2000}{200}$$

Dividing by 200 undoes the multiplication by 200.

$$n^{2}+\frac{600}{200}n=\frac{2000}{200}$$

Divide 600 by 200.

$$n^{2}+3n=\frac{2000}{200}$$

Divide 2000 by 200.

$$n^{2}+3n=10$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+3n+\left(\frac{3}{2}\right)^{2}=10+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+3n+\frac{9}{4}=10+\frac{9}{4}$$

Add 10 to \frac{9}{4}.

$$n^{2}+3n+\frac{9}{4}=\frac{49}{4}$$

Factor n^{2}+3n+\frac{9}{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}.

$$\left(n+\frac{3}{2}\right)^{2}=\frac{49}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{3}{2}\right)^{2}}=\sqrt{\frac{49}{4}}$$

Simplify.

$$n+\frac{3}{2}=\frac{7}{2} n+\frac{3}{2}=-\frac{7}{2}$$

Subtract \frac{3}{2} from both sides of the equation.

$$n=2 n=-5$$