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Choose the suitable method out of middle term splitting and factor theorem to factorize the following \[5{{p}^{2}}+17pq-12{{q}^{2}}\].

Last updated date: 20th Jul 2024
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Hint: In this problem, we have to find the factors of the given quadratic equation. We can take the common term outside the given quadratic equation. Then we will get a simplified quadratic equation, from which we can take the constant term, which is the result of multiplying two numbers and when adding/subtracting the same two numbers, we will get the coefficient of x, those two numbers are the factors of the equation.

Complete step by step answer:
We know that the given equation to be factored is,
We can now take the middle term 17pq, which can be split as,
\[\Rightarrow 17pq=20pq-3pq\]
We can now write the form after splitting the term, we get
\[\Rightarrow 5{{p}^{2}}+20pq-3pq-12{{q}^{2}}\]
We can now take common terms from the first two terms and the last two terms, we get
\[\Rightarrow 5p\left( p+4q \right)-3q\left( p+4q \right)\]
We can again take the common terms and write the remaining terms, we get
\[\Rightarrow \left( 5p-3q \right)\left( p+4q \right)\]
Therefore, the factors of the equation \[5{{p}^{2}}+17pq-12{{q}^{2}}\] are \[\left( 5p-3q \right)\left( p+4q \right)\].

Note: Students make mistakes while taking common terms from the given equation, which should be concentrated. We can also use a perfect square method to factorize the given equation or we can split the middle term into parts and take common terms from that to factorize the given equation. The number, which is taken as a common number outside in the first step, should be multiplied with the factors to get the correct answer.