Question

How do you factor $3{{x}^{2}}-4x-4$ ?

Answer 1

Hint: Now consider the given expression $3{{x}^{2}}-4x-4$ Now we will split the middle terms to solve the expression. The middle terms should be split in a way such that the product of the terms is equal to multiplication of the first term and last term. Now we will further simplify the expression and hence factorize the expression.

Complete step by step solution:
Now we are given with a quadratic expression in x.
We want to factor the given expression. To do so we will use splitting the middle terms method.
Now the middle terms of the expression is -4x.
Now we want to split the middle terms such that the multiplication of the two terms obtained after splitting is equal to the product of the first term and the last term.
Hence we will split the term -4x as -6x + 2x. Here $\left( -6x \right)\left( 2x \right)=-12{{x}^{2}}=3{{x}^{2}}\left( -4 \right)$
Hence we get the given expression as $3{{x}^{2}}-6x+2x-4$
Now let us take 3x common from the first two terms and 2 common from the last two terms.
Hence we get,
$\Rightarrow 3x\left( x-2 \right)+2\left( x-2 \right)$
Now taking x – 2 common from the whole expression we have $\left( x-2 \right)\left( 3x+2 \right)$ .
Now we cannot further factor the expression as we have linear expressions
Hence the given expression is factored.

Note: Now note that we can also use the roots to find the factors of the expression. Now if $\alpha $ and $\beta $ are the roots of the expression then $x-\alpha $ and $x-\beta $ are the factors of the expression. Now we know that the roots of the quadratic expression $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we can easily factor the given expression.