Question

How do you factor the expression $9{{x}^{2}}+6x+1$?

Answer 1

Hint: In the question, we need to find the factors of the equation $9{{x}^{2}}+6x+1$. Here to find the factors we need to know the type of the equation. The given equation has the variable with the exponent two, the equation will be quadratic. So we can solve the given equation factors in the method of simplifying the equation and using the method sum-product pattern.

Complete step-by-step answer:
$9{{x}^{2}}+6x+1$ Is the given equation in the question and we need to solve and find the factors.
As the given equation is in the form of quadratic. Factoring the factor will be easy.
Now let’s start with the first step. Here in the equation, we don’t have any negative sign for the highest exponent variable, so we can directly apply the sum-product pattern method.
In the given equation we need to divide one term into two-term without changing the value and this is done because we are applying the sum-product pattern.
Now we apply the method for the middle term in the given equation.
Now as we applying method the equation, it will be changed as
$\Rightarrow 9{{x}^{2}}+6x+1$
$\Rightarrow 9{{x}^{2}}+3x+3x+1$
Now the split equation has four terms.
We can find out the factors by taking the common factors from the pair terms.
$\Rightarrow 3x\left( 3x+1 \right)+3x+1$
Now we need to write
$\Rightarrow \left( 3x+1 \right)\left( 3x+1 \right)$
Now take the term
$\Rightarrow 3x+1=0$
$\begin{align}
  & \Rightarrow 3x=-1 \\
 & \Rightarrow x=\dfrac{-1}{3} \\
\end{align}$
As the two terms are $\left( 3x+1 \right)$.
Therefore the factors for the equation $9{{x}^{2}}+6x+1$ are $\dfrac{-1}{3},\dfrac{-1}{3}$

Note: Here in the equation there is no negative sign so, we solved the equation directly. If the equation has a negative sign for the term with the highest exponent then we need to separate the negative sign and write the positive equation in the bracket and solve the equation in the bracket.