Question

How do you factor $3{{x}^{2}}+9x-12=0$?

Answer 1

Now the given expression is a quadratic expression in x. Now to factorize the expression we will split the middle terms such that when we take the product of the two terms obtained after splitting we will get the product of the first term and the last term. Now we will simplify the obtained expression and hence we have the factorization of the given expression.

Complete step by step solution:
Now consider the given equation $3{{x}^{2}}+9x-12=0$ .
Now we know that the above equation is a quadratic equation in x.
Now we want to find the factors of the expression and hence express the equation in terms of product of factors.
Now to factorize the expression we will split the middle terms of the expression such that the multiplication of the terms is the product of the first terms and the last term.
Hence let us write 9x as 12x – 3x. as $\left( 12x \right)\left( -3x \right)=-36{{x}^{2}}=3{{x}^{2}}\times \left( -12 \right)$ .
Hence we get the expression as,
$\Rightarrow 3{{x}^{2}}+12x-3x-12=0$
Now taking $3x$ common from the first two terms and -3 common from the last two terms we get the expression as,
$\Rightarrow 3x\left( x+4 \right)-3\left( x+4 \right)$
Now again taking x+4 common from the expression we get,
$\Rightarrow \left( 3x-3 \right)\left( x+4 \right)$
Hence we have $3{{x}^{2}}+9x-12=\left( 3x-3 \right)\left( x+4 \right)$ .
Hence we have factorized the given expression.

Note: Now note that we can also find the roots of the given quadratic expression. Now we know that the roots of quadratic expression of the form $a{{x}^{2}}+bx+c$ is given by the formula $\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a}$ . Hence we will first calculate the roots of the expression. Now the factors of the expression will be given by $\left( x-\dfrac{-b\pm \sqrt{{{b}^{2}}-4ac}}{2a} \right)$ . Hence we have the factorization of the given expression.