Solve 200=-3n^2+49n

$$200 = - 3 n ^ { 2 } + 49 n$$

Solve for n

$$n=8$$

$$n = \frac{25}{3} = 8\frac{1}{3} \approx 8.333333333$$

Steps Using Factoring By Grouping

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

$$-3n^{2}+49n=200$$

Subtract 200 from both sides.

$$-3n^{2}+49n-200=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as -3n^{2}+an+bn-200. To find a and b, set up a system to be solved.

$$a+b=49 ab=-3\left(-200\right)=600$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 600.

$$1,600 2,300 3,200 4,150 5,120 6,100 8,75 10,60 12,50 15,40 20,30 24,25$$

Calculate the sum for each pair.

$$1+600=601 2+300=302 3+200=203 4+150=154 5+120=125 6+100=106 8+75=83 10+60=70 12+50=62 15+40=55 20+30=50 24+25=49$$

The solution is the pair that gives sum 49.

$$a=25 b=24$$

Rewrite -3n^{2}+49n-200 as \left(-3n^{2}+25n\right)+\left(24n-200\right).

$$\left(-3n^{2}+25n\right)+\left(24n-200\right)$$

Factor out -n in the first and 8 in the second group.

$$-n\left(3n-25\right)+8\left(3n-25\right)$$

Factor out common term 3n-25 by using distributive property.

$$\left(3n-25\right)\left(-n+8\right)$$

To find equation solutions, solve 3n-25=0 and -n+8=0.

$$n=\frac{25}{3} n=8$$

Steps Using the Quadratic Formula

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

$$-3n^{2}+49n=200$$

Subtract 200 from both sides.

$$-3n^{2}+49n-200=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 49 for b, and -200 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$n=\frac{-49±\sqrt{49^{2}-4\left(-3\right)\left(-200\right)}}{2\left(-3\right)}$$

Square 49.

$$n=\frac{-49±\sqrt{2401-4\left(-3\right)\left(-200\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$n=\frac{-49±\sqrt{2401+12\left(-200\right)}}{2\left(-3\right)}$$

Multiply 12 times -200.

$$n=\frac{-49±\sqrt{2401-2400}}{2\left(-3\right)}$$

Add 2401 to -2400.

$$n=\frac{-49±\sqrt{1}}{2\left(-3\right)}$$

Take the square root of 1.

$$n=\frac{-49±1}{2\left(-3\right)}$$

Multiply 2 times -3.

$$n=\frac{-49±1}{-6}$$

Now solve the equation n=\frac{-49±1}{-6} when ± is plus. Add -49 to 1.

$$n=-\frac{48}{-6}$$

Divide -48 by -6.

$$n=8$$

Now solve the equation n=\frac{-49±1}{-6} when ± is minus. Subtract 1 from -49.

$$n=-\frac{50}{-6}$$

Reduce the fraction \frac{-50}{-6} to lowest terms by extracting and canceling out 2.

$$n=\frac{25}{3}$$

The equation is now solved.

$$n=8 n=\frac{25}{3}$$

Steps for Completing the Square

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

$$-3n^{2}+49n=200$$

Divide both sides by -3.

$$\frac{-3n^{2}+49n}{-3}=\frac{200}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$n^{2}+\frac{49}{-3}n=\frac{200}{-3}$$

Divide 49 by -3.

$$n^{2}-\frac{49}{3}n=\frac{200}{-3}$$

Divide 200 by -3.

$$n^{2}-\frac{49}{3}n=-\frac{200}{3}$$

Divide -\frac{49}{3}, the coefficient of the x term, by 2 to get -\frac{49}{6}. Then add the square of -\frac{49}{6} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$n^{2}-\frac{49}{3}n+\left(-\frac{49}{6}\right)^{2}=-\frac{200}{3}+\left(-\frac{49}{6}\right)^{2}$$

Square -\frac{49}{6} by squaring both the numerator and the denominator of the fraction.

$$n^{2}-\frac{49}{3}n+\frac{2401}{36}=-\frac{200}{3}+\frac{2401}{36}$$

Add -\frac{200}{3} to \frac{2401}{36} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$n^{2}-\frac{49}{3}n+\frac{2401}{36}=\frac{1}{36}$$

Factor n^{2}-\frac{49}{3}n+\frac{2401}{36}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(n-\frac{49}{6}\right)^{2}=\frac{1}{36}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(n-\frac{49}{6}\right)^{2}}=\sqrt{\frac{1}{36}}$$

Simplify.

$$n-\frac{49}{6}=\frac{1}{6} n-\frac{49}{6}=-\frac{1}{6}$$

Add \frac{49}{6} to both sides of the equation.

$$n=\frac{25}{3} n=8$$