Question

How do you factor ${{y}^{2}}+2xy-63{{x}^{2}}$ ?

Answer 1

Hint: This type of question can be solved simply by splitting the terms as sum of two terms and product of two terms. We then need to take the common terms out and convert it into product form thereby factoring it.

Complete step-by-step answer:
Question given is ${{y}^{2}}+2xy-63{{x}^{2}}$ which is a polynomial equation that needs to be factorized. It has a ${{y}^{2}}$ term that has a coefficient of 1. It has a ${{x}^{2}}$ term that has a coefficient of -63. We also have a $xy$ term with a coefficient of +2.
So, firstly we have to split the +2 term as a sum of two terms say $a$ and $b$ such that,
$a+b=+2......\left( 1 \right)$
Now let us consider the -63 term which can be represented as a product of the same two terms $a$ and $b$ such that,
$a.b=-63......\left( 2 \right)$
From the multiplication tables, we can see many combinations for 63 such as by multiplying -21 by 3, 21 by -3, -9 by 7 and 9 by -7. But from this set, we need to pick the numbers such that the sum of the two is equal to +2. We see only the combination of 9 and -7 satisfies this condition.
Therefore, by using this in the question,
$\Rightarrow {{y}^{2}}+9xy-7xy+(9.-7){{x}^{2}}$
$\Rightarrow {{y}^{2}}+9xy-7xy-9.7{{x}^{2}}$
Taking out $y$ common from first two terms and $-7x$ common from next 2 terms,
$\Rightarrow y(y+9x)-7x(y+9x)$
Taking the term $(y+9x)$ out common we get,
$\Rightarrow (y+9x).(y-7x)......(3)$
Therefore, we have split or factorized the above question into the equation given above.
Hence the final answer is $(y+9x).(y-7x)$ .

Note: While solving this question, the students need to be careful while choosing the factors. This step can be performed well if students have a fair and sound knowledge of their multiplication tables. Also, care must be taken while noting down the sign before splitting the number because a wrong sign can lead to selection of wrong factors and the solution itself will be wrong.