Question

How do you factor \[4{x^2} - 12x + 9\]?

Answer 1

Hint: We will use the method of splitting the middle term to factorize the given polynomial. In order to do this, we will write the middle term as a sum of two terms. Then, we will club together like terms and take common factors out. Finally, we will get the polynomial as a product of two factors.

Complete step by step solution:
The quadratic polynomial given to us is \[4{x^2} - 12x + 9\].
We have to factorize this polynomial i.e.; we have to find two factors such that the given polynomial can be expressed as a product of the two factors. To do this, we will use the method of splitting the middle term.
In the given polynomial \[4{x^2} - 12x + 9\], we have to split \[ - 12\] as a sum of two terms whose product is \[4 \times 9 = 36\].
Let us find the factors of 36 and find combinations of numbers that in addition or subtraction will give us \[ - 12\].
We know that \[36 = 2 \times 2 \times 3 \times 3\].
We can also write this as \[36 = 6 \times 6\].
We also know that \[ - 12 = ( - 6) + ( - 6)\].
So, the required numbers are \[ - 6\] and \[ - 6\].
Hence, the polynomial can be written as
\[4{x^2} - 12x + 9 = 4{x^2} - 6x + ( - 6x) + 9\]
We will club the first two terms and the last two terms together.
In the first two terms, the common factor is \[2x\]. In the last two terms, the common factor is \[ - 3\]. Thus, we get
\[ \Rightarrow 4{x^2} - 12x + 9 = 2x(2x - 3) - 3(2x - 3)\]
We can see on the RHS that the factor \[(2x - 3)\] is common to both terms.
Factoring out the common terms, we get:

\[ \Rightarrow 4{x^2} - 12x + 9 = (2x - 3)(2x - 3)\]
\[ \Rightarrow 4{x^2} - 12x + 9 = {(2x - 3)^2}\]

Therefore, we have expressed the given polynomial as a product of two factors.

Note: To factorize a polynomial \[a{x^2} + bx + c\] by splitting the middle term, we have to find two terms such that we can write \[b\] as a sum of the two terms such that their product is \[a \times c\]. This means that we find two numbers \[p\] and \[q\] such that \[p + q = b\] and \[pq = ac\]. After finding \[p\] and \[q\], we split the middle term in the quadratic polynomial as \[px + qx\] and get desired factors by grouping the terms. Here, the terms \[p\] and \[q\] are not necessarily positive terms.