Question

How do you factor$42{x^2} + 35x + 7$?

Answer 1

Hint: In the given question, we have been asked to find factors of a quadratic equation with a single variable. Quadratic equations are the polynomial expressions which have one variable with degree 2. Since in the equation variable has degree 2, $x$ will have two values. Factoring of a linear equation is basically representing it in the form $(x - \alpha )(x - \beta )$. In order to proceed with the following question we need to use splitting the middle term method.

Complete step by step solution:
We are given,
$ \Rightarrow 42{x^2} + 35x + 7$
Since we know the equation is in standard form, we can solve the equation by splitting the middle term.
To solve the splitting method, we need to find the factors of the number obtained by multiplying the coefficient of ${x^2}$ i.e. $42$ by constant term i.e. $7$. The number we’ll obtain is $294$. We’ll have to find the factors of $294$ whose sum is equal to the coefficient of the middle term, which is $35$.
Rewrite the equation by splitting the middle term using those two factors.
\[ \Rightarrow 42{x^2} + 21x + 14x + 7\]
Now, we have to pull out the common factors from 1st two terms
\[ \Rightarrow 21x(2x + 1)\]
and last two terms
\[ \Rightarrow 7(2x + 1)\]
After adding up the four terms we’ll get
\[ \Rightarrow (3x + 7)(2x + 1)\]
These are the required factors of the above expression.

Note: Before solving any question of quadratic equation, ensure that the equation is of the form $a{x^2} + bx + c = 0$, and if it is not then convert it in this form, where $a,b,c \in R$ and $a \ne 0$. The factors obtained in the splitting method should be a co-prime number, i.e. only one positive number should divide them both.