Question

How do you factor the trinomial $ 4{x^2} + 12x + 9 $ ?

Answer 1

Hint: In this equation, we need to find the factors of the trinomial expression. If you look at the expression carefully you will come to know that the expression is of the form complete squares. It will have two equal factors.

Complete step-by-step answer:
In this question, we had a trinomial expression and we had to find the factors for this expression. Trinomial is the math equation having three terms which are connected through the plus or minus notations.
The given expression is $ 4{x^2} + 12x + 9 $ a quadratic equation.
Finding the factors of the quadratic equation with the help of factoring method. For that we need to assume two numbers which has the sum is equal to negative of the ratio of coefficient of $ x $ to the coefficient of $ {x^2} $ and the product is equal to the ratio of constant term to the coefficient of $ {x^2} $
Let the quadratic equation be $ a{x^2} + bx + c = 0 $
Sum = $ \dfrac{{ - b}}{a} $
Product = $ \dfrac{c}{a} $
Then, the two such numbers are $ 6,6 $
Splitting the middle term by these two numbers,
 $\Rightarrow 4{x^2} + 6x + 6x + 9 = 0 $
Taking common from first two and last two terms respectively.
 $ 2x(2x + 3) + 3(2x + 3) = 0 $
Combing the two factors
 $ \left( {2x + 3} \right)\left( {2x + 3} \right) = 0 $
Equating to zero
 $\Rightarrow x = - \dfrac{3}{2}, - \dfrac{3}{2} $
This is our required solution.
So, the correct answer is “ $ x = - \dfrac{3}{2}, - \dfrac{3}{2} $ ”.

Note: The factors can be found by many methods. It is not compulsory to follow this method only. If we noticed that the expression is a type of complete squares, so the two factors are found to be equal. If we come to know that the expression is complete squares it will solve the question in hardly two steps.