Question

Find the roots of the equation: \[25{{a}^{2}}-35a+12\]
A) \[(5a-3)(5a-4)\]
B) \[(5a-3)(5a+4)\]
C) \[(5a+3)(5a-4)\]
D) \[(5a+3)(5a+4)\]

Answer 1

Hint: Use Lowest common multiple to find the roots. To find the roots we will use middle term form where we split the middle term. Here in the equation \[25{{a}^{2}}-35a+12\] the middle term is \[35a\].

Complete step by step answer:

We need to find the lowest common multiple of 25 and 12 from the first and last term respectively. We find the lowest common factor of these two numbers because in the end we need to have a common multiple from these two terms and the middle term. So instead of breaking the middle term one by one and checking if it has a common multiple with the first and last term we do this step where we find the Lowest common multiple.
Therefore, the next steps will be,
\[\begin{gathered}
& 12\times 25=300 \\
& LCM\text{ of }300=2\times 2\times 3\times 5\times 5 \\
\end{gathered}\]
Now that we have so many factors in this lowest common multiple we can use it to check if by adding or subtracting these factors we can end up getting 35 from the middle term….
\[\begin{gathered}
& 2\times 2=4 \\
& 3\times 5\times 5=45 \\
& 45+4=49 \\
& 45-4=41 \\
\end{gathered}\]
Neither of these two gives 35 so we on and proceed with;
\[\begin{gathered}
& 2\times 2\times 3=12 \\
& 5\times 5=25 \\
& 12+25=37 \\
& 25-12=13 \\
\end{gathered}\]
Again, nothing we got here, so next step;
\[\begin{gathered}
& 2\times 2\times 5=20 \\
& 5\times 3=15 \\
& 20+15=35 \\
\end{gathered}\]
So in this step we got the 35. Therefore we need to split 35 into 20 and 15. Thus next will be,
\[\begin{gathered}
& 25{{a}^{2}}-20a-15a+12 \\
& 5a(5a-4)-3(5a-4)\text{ }\!\![\!\!\text{ common multiple 5 , 3 and 4 }\!\!]\!\!\text{ } \\
& (5a-3)(5a-4) \\
\end{gathered}\]

Thus, the answer of this question will be option A \[(5a-3)(5a-4)\].

Note: Multiplying the first and last term is necessary since we need both the terms to get a common multiple properly.