Question

How do you factor the expression \[3{{x}^{2}}+12x+12=0\]?

Answer 1

Hint: In this problem, we have to find the factor of the given expression \[3{{x}^{2}}+12x+12=0\] by factorization method. We can first take the middle term for splitting into two terms whose addition is itself and multiplication is equal to \[12\times 3=36=6\times 6\], which is the multiplication of the first term and the last term. We can then take common terms outside to get the factors.

Complete step by step solution:
We know that the given expression is,
\[3{{x}^{2}}+12x+12=0\]
We can first split the middle term to form factors.
We have to expand the middle term i.e. the x term with its coefficient in such a way that their addition is equal to the middle term i.e. 12x, and multiplication is equal to \[12\times 3=36=6\times 6\]
\[\begin{align}
  & \Rightarrow 12\times 3=36=6\times 6 \\
 & \Rightarrow 6+6=12 \\
\end{align}\]
We can apply the above step,
\[3{{x}^{2}}+6x+6x+12=0\]
We can now take the first two terms and the last two terms to take common terms outside, we get
\[\Rightarrow \left( 3{{x}^{2}}+6x \right)+\left( 6x+12 \right)\]
Now we can take the common terms outside, we get
\[\Rightarrow 3x\left( x+2 \right)+6\left( x+2 \right)\]
We can again take the common factor first then the remaining terms to make a factor, we get
\[\Rightarrow \left( 3x+6 \right)\left( x+2 \right)\]
Therefore, the factors are \[\left( 3x+6 \right)\left( x+2 \right)\].

Note: Students make mistakes while taking common factors from the equation to form the factors of the equation in which we should concentrate. We can also verify for the correct answer using the quadratic formula by solving for x.
The quadratic formula for the equation \[a{{x}^{2}}+bx+c\] is
\[x=\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}\]
 We know that the given expression is,
\[3{{x}^{2}}+12x+12=0\]
Where, a = 3, b = 12, c = 12.
We can substitute the above values in quadratic formula.
\[\begin{align}
  & \Rightarrow x=\dfrac{-12\pm \sqrt{144-144}}{98} \\
 & \Rightarrow x=\dfrac{-12}{98} \\
 & \Rightarrow x=\dfrac{-6}{3}=-2 \\
\end{align}\]
Where,\[x=-\dfrac{6}{3},x=-2\]
We can take the term in the right-hand side, to the left-hand side by changing its sign.
\[\Rightarrow \left( 3x+6 \right)\left( x+2 \right)\]
Therefore, the factors are \[\left( 3x+6 \right)\left( x+2 \right)\].