Question

How do you factor \[9{{x}^{2}}+30xy+25{{y}^{2}}\] ?

Answer 1

Hint: From the question given equation is \[9{{x}^{2}}+30xy+25{{y}^{2}}\].For answering this question we will use factorization. Factorization is the process of deriving factors of a number which divides the given number evenly. Factorization is the process of reducing the bracket of a reducing the bracket of a quadratic equation, instead of expanding the bracket and converting the equation to a product of factors which cannot be reduced further. There are many methods for the factorization process. Now, we will do the given question by the method of splitting the constant and doing the sum product pattern.

Complete step by step solution:
Now considering from the question we have an expression \[9{{x}^{2}}+30xy+25{{y}^{2}}\] for which we need to derive the factors.
We can factor the \[9{{x}^{2}}+30xy+25{{y}^{2}}\] by below method:
Given equation is in the form of \[a{{x}^{2}}+bx+c=0\]
First, we have to divide \[225\] into the product of the two numbers and the Sum of those two numbers must be equal to the coefficient of\[x\].
Now, \[225\]can be split into the product of the two numbers.
\[225\]can be split into products of \[15\]and \[15\].
Their sum is also equal to \[30\]which is equal to the coefficient of\[x\].
\[9{{x}^{2}}+30xy+25{{y}^{2}}\]
\[\Rightarrow 9{{x}^{2}}+15xy+15xy+25{{y}^{2}}=0\]
\[\Rightarrow 3x\left( 3x+5y \right)+5y\left( 3x+5y \right)=0\]
\[\Rightarrow \left( 3x+5y \right)\left( 3x+5y \right)=0\]
Therefore, \[\left( 3x+5y \right)\] ,\[\left( 3x+5y \right)\] is the factors of \[9{{x}^{2}}+30xy+25{{y}^{2}}\]

Note:
During answering questions of this type students can recognise it as a perfect square trinomial. In general, \[{{\left( a\pm b \right)}^{2}}={{a}^{2}}\pm 2ab+{{b}^{2}}\]
In the above question student can notice that \[9{{x}^{2}}={{\left( 3x \right)}^{2}}\] and \[25{{y}^{2}}={{\left( 5y \right)}^{2}}\] are both perfect squares, so the question is if we let \[a=3x\] and \[b=5y\] then does \[\pm 2ab\] match the middle term \[30xy\] with a positive sign. Hence: \[{{\left( 3x+5y \right)}^{2}}=9{{x}^{2}}+30xy+25{{y}^{2}}\]