Question

How do you find the factors of \[5{{x}^{2}}-17x+6\]?

Answer 1

Hint: This type of question is based on the concept of factorisation. We can solve this question by splitting the middle term in such a way that it has common terms from the first and last term. Here -17x=-15x-2x. Substitute this in the obtained equation. Then, make some necessary calculations and take common terms outside the bracket. Thus, we get the factors of the given equation \[5{{x}^{2}}-17x+6\].

Complete answer:
According to the question, we are asked to find the factors of the given equation\[5{{x}^{2}}-17x+6\].
We have been given the equation is \[5{{x}^{2}}-17x+6\]. ------(1)
Let us first split the middle term of the equation (1).
We know that -17x=-15x-2x.
Substituting this in equation (1), we get,
\[5{{x}^{2}}-17x+6=5{{x}^{2}}-15x-2x+6\]
On further simplification, we get,
\[\Rightarrow 5{{x}^{2}}-17x+6=5{{x}^{2}}-5\times 3x-2x+3\times 2\]
Now, let us take 5x common from the first two terms and -2 common from the last two terms.
\[\Rightarrow 5{{x}^{2}}-17x+6=5x\left( x-3 \right)-2\left( x-3 \right)\] ---------(2)
 Let us take \[\left( x-3 \right)\] common from equation (2).
\[5{{x}^{2}}-17x+6=\left( x-3 \right)\left( 5x-2 \right)\]
Whenever we convert a function into a product of two functions, there are considered to be the factors of the function.
Therefore, the factors are \[\left( x-3 \right)\] and \[\left( 5x-2 \right)\].
Hence, the factors of \[5{{x}^{2}}-17x+6\] are \[\left( x-3 \right)\] and \[\left( 5x-2 \right)\].

Note: Whenever you get this type of problem, we should always try to make the necessary calculations in the given equation to get the final of x. We should then find the factors as required. Then, we should always split the middle terms in such a way that we get common terms from the given equation. We should also avoid calculation mistakes based on sign conventions. Similarly, we can find the factors of \[{{x}^{2}}-11x-10\].