Question

How do you factor $ 15{m^2} - 25m $ ?

Answer 1

Hint: Take the highest factors common from both the operands. For example, $ 15 $ and $ 25 $ have the highest factor $ 5 $ common in them, whereas in $ {m^2} $ and $ m $ ,the highest factor common is $ m $. Just take the common values out and simplify until we get the terms which cannot be further splitted or simplified.

Complete step by step solution:
Let $ 15{m^2} - 25m $ be represented as $ f(m) $ .
 $ f(m) = 15{m^2} - 25m $
Take the highest common factors from the operands.
Factors of $ 15 = 1,3,5,15 $
Factors of $ 25 = 1,5,25 $
Factors of $ {m^2} = 1,m,{m^2} $
Factors of $ m = 1,m $
In $ 15 $ and $ 25 $ ,we have $ 5 $ as a highest factor.
And in $ {m^2} $ and $ m $ , we have $ m $ as a highest factor.
Now, from $ f(m) = 15{m^2} - 25m $ , take the common factors out and we get:
 $
  f(m) = (3 \times 5 \times m \times m - 5 \times 5 \times m) \\
  f(m) = 5m(3m - 5) \;
  $
Since, we can see that the values cannot be further divided, so $ 5m $ and $ (3m - 5) $ are the factors of \[f(m)\].
Put
\[
  f(m) = 0 \\
  5m(3m - 5) = 0 \;
 \]
So, either $ 5m = 0 $ or $ (3m - 5) = 0 $
So, $ m = 0 $ and $ m = \dfrac{5}{3} $ are the factors of $ f(m) $ .
Check:
Let’s put $ m = \dfrac{5}{3} $ in $ f(m) $ to check whether it gives the required output or not:
 $
  f(m) = 5m(3m - 5) \\
  f(\dfrac{5}{3}) = 5 \times \dfrac{5}{3}(3 \times \dfrac{5}{3} - 5) \\
   = \dfrac{{25}}{3}(5 - 5) \\
   = \dfrac{{25}}{3} \times 0 \\
   = 0 \;
  $

Therefore, the factors are correct.
Overall, it can be written as: The factors of\[f(m)\]are $ 5m $ and $ (3m - 5) $ .
Or, \[f(0)\] and \[f(\dfrac{5}{3})\] are the factors of \[f(m)\].
Alternate method:
You can also use the Quadratic Equation method to solve this, since it is a Quadratic Equation in the form $ a{x^2} + bx + c $ .
Formula of Quadratic equation is given as: $ x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}} $ . Just put the values and get the results.

Note: Check the common factors first and take highest factors as common factors as always to avoid further simplification. If there are no common factors among the operands then leave it as it or can try some other methods given to solve.