Solve 50m^2-225m+63

$$50 { m }^{ 2 } -225m+63$$

Factor

$$\left(5m-21\right)\left(10m-3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 50m^{2}+am+bm+63. To find a and b, set up a system to be solved.

$$a+b=-225 ab=50\times 63=3150$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 3150.

$$-1,-3150 -2,-1575 -3,-1050 -5,-630 -6,-525 -7,-450 -9,-350 -10,-315 -14,-225 -15,-210 -18,-175 -21,-150 -25,-126 -30,-105 -35,-90 -42,-75 -45,-70 -50,-63$$

Calculate the sum for each pair.

$$-1-3150=-3151 -2-1575=-1577 -3-1050=-1053 -5-630=-635 -6-525=-531 -7-450=-457 -9-350=-359 -10-315=-325 -14-225=-239 -15-210=-225 -18-175=-193 -21-150=-171 -25-126=-151 -30-105=-135 -35-90=-125 -42-75=-117 -45-70=-115 -50-63=-113$$

The solution is the pair that gives sum -225.

$$a=-210 b=-15$$

Rewrite 50m^{2}-225m+63 as \left(50m^{2}-210m\right)+\left(-15m+63\right).

$$\left(50m^{2}-210m\right)+\left(-15m+63\right)$$

Factor out 10m in the first and -3 in the second group.

$$10m\left(5m-21\right)-3\left(5m-21\right)$$

Factor out common term 5m-21 by using distributive property.

$$\left(5m-21\right)\left(10m-3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$50m^{2}-225m+63=0$$

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.

$$m=\frac{-\left(-225\right)±\sqrt{\left(-225\right)^{2}-4\times 50\times 63}}{2\times 50}$$

Square -225.

$$m=\frac{-\left(-225\right)±\sqrt{50625-4\times 50\times 63}}{2\times 50}$$

Multiply -4 times 50.

$$m=\frac{-\left(-225\right)±\sqrt{50625-200\times 63}}{2\times 50}$$

Multiply -200 times 63.

$$m=\frac{-\left(-225\right)±\sqrt{50625-12600}}{2\times 50}$$

Add 50625 to -12600.

$$m=\frac{-\left(-225\right)±\sqrt{38025}}{2\times 50}$$

Take the square root of 38025.

$$m=\frac{-\left(-225\right)±195}{2\times 50}$$

The opposite of -225 is 225.

$$m=\frac{225±195}{2\times 50}$$

Multiply 2 times 50.

$$m=\frac{225±195}{100}$$

Now solve the equation m=\frac{225±195}{100} when ± is plus. Add 225 to 195.

$$m=\frac{420}{100}$$

Reduce the fraction \frac{420}{100} to lowest terms by extracting and canceling out 20.

$$m=\frac{21}{5}$$

Now solve the equation m=\frac{225±195}{100} when ± is minus. Subtract 195 from 225.

$$m=\frac{30}{100}$$

Reduce the fraction \frac{30}{100} to lowest terms by extracting and canceling out 10.

$$m=\frac{3}{10}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{21}{5} for x_{1} and \frac{3}{10} for x_{2}.

$$50m^{2}-225m+63=50\left(m-\frac{21}{5}\right)\left(m-\frac{3}{10}\right)$$

Subtract \frac{21}{5} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$50m^{2}-225m+63=50\times \frac{5m-21}{5}\left(m-\frac{3}{10}\right)$$

Subtract \frac{3}{10} from m by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$50m^{2}-225m+63=50\times \frac{5m-21}{5}\times \frac{10m-3}{10}$$

Multiply \frac{5m-21}{5} times \frac{10m-3}{10} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$50m^{2}-225m+63=50\times \frac{\left(5m-21\right)\left(10m-3\right)}{5\times 10}$$

Multiply 5 times 10.

$$50m^{2}-225m+63=50\times \frac{\left(5m-21\right)\left(10m-3\right)}{50}$$

Cancel out 50, the greatest common factor in 50 and 50.

$$50m^{2}-225m+63=\left(5m-21\right)\left(10m-3\right)$$

Evaluate

$$\left(5m-21\right)\left(10m-3\right)$$