Question

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

Answer 1

Hint: When we have a polynomial of the form \[ax{{}^{2}}+bx+c\], we can factor this quadratic by splitting up the b term into two terms based on the signs of a and c terms. If we have the same sign for both a and c terms, then we split the b term into two parts. These parts are formed such that the sum of them is b term itself and their product is the same as that of the product of a and c terms.

Complete step by step answer:
In the given quadratic polynomial \[9x{{}^{2}}+12x+4\], the coefficient of \[x{{}^{2}}\] and constant terms are of the same sign and their product is 36. That is, we have a and c coefficients with the same sign in the polynomial of form \[ax{{}^{2}}+bx+c\].
Hence, we would split 12, which is the coefficient of x, into two parts, whose sum is 12 and the product is 36. These are 6 and 6.
So, we write it as
\[\Rightarrow 9{{x}^{2}}+12x+4\]
We now split 12x into 6x and 6x
\[\Rightarrow 9{{x}^{2}}+6x+6x+4\]
We take the terms common in first 2 terms and last 2 terms
\[\Rightarrow 3x(3x+2)+2(3x+2)\]
Here, we have \[(3x+2)\] in common then
\[\Rightarrow (3x+2)(3x+2)={{(3x+2)}^{2}}\]
\[\therefore \] \[9x{{}^{2}}+12x+4={{(3x+2)}^{2}}\] is the required answer.

Note:
 In the given equation, the coefficients a and c are perfect squares. So, the equation can be written as \[(3x){{}^{2}}+2(3x)(2)+{{2}^{2}}\]. It is in the form of \[{{a}^{2}}+2ab+{{b}^{2}}\]. We know that \[{{(a+b)}^{2}}={{a}^{2}}+2ab+{{b}^{2}}\]. Therefore, \[{{(3x)}^{2}}+2(3x)(2)+{{2}^{2}}={{(3x+2)}^{2}}\]. If a and c are not perfect squares and b term cannot be split such that sum of those parts is b term and product is the same as that of the product of a and c terms. In such cases, we better go with the quadratic formula.