Solve 9800n+200n^2=1400

$$9800n+200 { n }^{ 2 } = 1400$$

Solve for n

$$n=\frac{\sqrt{2429}-49}{2}\approx 0.142443061$$

$$n=\frac{-\sqrt{2429}-49}{2}\approx -49.142443061$$

Steps Using the Quadratic Formula

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.

$$200n^{2}+9800n=1400$$

Subtract 1400 from both sides of the equation.

$$200n^{2}+9800n-1400=1400-1400$$

Subtracting 1400 from itself leaves 0.

$$200n^{2}+9800n-1400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 200 for a, 9800 for b, and -1400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-9800±\sqrt{9800^{2}-4\times 200\left(-1400\right)}}{2\times 200}$$

Square 9800.

$$n=\frac{-9800±\sqrt{96040000-4\times 200\left(-1400\right)}}{2\times 200}$$

Multiply -4 times 200.

$$n=\frac{-9800±\sqrt{96040000-800\left(-1400\right)}}{2\times 200}$$

Multiply -800 times -1400.

$$n=\frac{-9800±\sqrt{96040000+1120000}}{2\times 200}$$

Add 96040000 to 1120000.

$$n=\frac{-9800±\sqrt{97160000}}{2\times 200}$$

Take the square root of 97160000.

$$n=\frac{-9800±200\sqrt{2429}}{2\times 200}$$

Multiply 2 times 200.

$$n=\frac{-9800±200\sqrt{2429}}{400}$$

Now solve the equation n=\frac{-9800±200\sqrt{2429}}{400} when ± is plus. Add -9800 to 200\sqrt{2429}.

$$n=\frac{200\sqrt{2429}-9800}{400}$$

Divide -9800+200\sqrt{2429} by 400.

$$n=\frac{\sqrt{2429}-49}{2}$$

Now solve the equation n=\frac{-9800±200\sqrt{2429}}{400} when ± is minus. Subtract 200\sqrt{2429} from -9800.

$$n=\frac{-200\sqrt{2429}-9800}{400}$$

Divide -9800-200\sqrt{2429} by 400.

$$n=\frac{-\sqrt{2429}-49}{2}$$

The equation is now solved.

$$n=\frac{\sqrt{2429}-49}{2} n=\frac{-\sqrt{2429}-49}{2}$$

Steps for Completing the Square

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.

$$200n^{2}+9800n=1400$$

Divide both sides by 200.

$$\frac{200n^{2}+9800n}{200}=\frac{1400}{200}$$

Dividing by 200 undoes the multiplication by 200.

$$n^{2}+\frac{9800}{200}n=\frac{1400}{200}$$

Divide 9800 by 200.

$$n^{2}+49n=\frac{1400}{200}$$

Divide 1400 by 200.

$$n^{2}+49n=7$$

Divide 49, the coefficient of the x term, by 2 to get \frac{49}{2}. Then add the square of \frac{49}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}+49n+\left(\frac{49}{2}\right)^{2}=7+\left(\frac{49}{2}\right)^{2}$$

Square \frac{49}{2} by squaring both the numerator and the denominator of the fraction.

$$n^{2}+49n+\frac{2401}{4}=7+\frac{2401}{4}$$

Add 7 to \frac{2401}{4}.

$$n^{2}+49n+\frac{2401}{4}=\frac{2429}{4}$$

Factor n^{2}+49n+\frac{2401}{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}.

$$\left(n+\frac{49}{2}\right)^{2}=\frac{2429}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n+\frac{49}{2}\right)^{2}}=\sqrt{\frac{2429}{4}}$$

Simplify.

$$n+\frac{49}{2}=\frac{\sqrt{2429}}{2} n+\frac{49}{2}=-\frac{\sqrt{2429}}{2}$$

Subtract \frac{49}{2} from both sides of the equation.

$$n=\frac{\sqrt{2429}-49}{2} n=\frac{-\sqrt{2429}-49}{2}$$