Question

How do you solve \[3{{x}^{2}}+5x+2=0\] by factoring?

Answer 1

Hint: we can do it in different ways. But here we will use factoring or grouping methods to solve the equation. First we have to split our middle to satisfy the constraints we have in the method. Once we have splitted the middle term we have to rearrange the terms by taking common variables or numerical to arrive at our solution.

Complete step-by-step solution:
The rules to split the middle term are
1.The sum of the two terms is equal to middle term and
2.The product must be equal to the product of first and last terms.
So we have to make sure to satisfy these conditions while splitting the middle term.
Given equation is
\[3{{x}^{2}}+5x+2=0\]
Here to split the middle term the easy way to find the factors is to do prime factorization of the product of the first and last terms.
So in our case we have found the prime factors for \[6\].
The prime factors for \[6\] are
\[6=2\times 3\]
From the prime factors we have to take two factors through which we will get \[6\] and \[5\].
We can see that
\[6=2\times 3\]
\[5=2+3\]
So we can split our middle term into \[2\] and \[3\]
Then the equation will look like
\[\Rightarrow 3{{x}^{2}}+3x+2x+2=0\]
Now we can rearrange the terms by taking common terms
From the above equation we can take \[3x\] as common in first two terms and \[2\] as common in next two terms
Then the equation will become
\[\Rightarrow 3x\left( x+1 \right)+2\left( x+1 \right)=0\]
Now we can take \[3x+2\] as common in two terms we will get
\[\Rightarrow \left( x+1 \right)\left( 3x+2 \right)=0\]
So the factors of the equation \[{{x}^{2}}-2x-24\] are \[\left( x+1 \right)\left( 3x+2 \right)\].
Now we have to solve for x values we will get
\[\Rightarrow x+1=0\]
\[\Rightarrow 3x+2=0\]
\[\Rightarrow x=-2\]
\[\Rightarrow 3x=-2\]
\[\Rightarrow x=\dfrac{-2}{3}\]
So the values of x are \[\Rightarrow x=-1,\dfrac{-2}{3}\].

Note: We can also solve the problem using quadratic formulas also. In this method we have to find the discriminant and substitute the values in the formula we have. But here it is clearly mentioned to use factoring methods.