Question

How do you factor $2{{x}^{2}}+4x-6$?

Answer 1

Hint: Now the given expression is a quadratic expression in x. We will factorize the expression using splitting the middle term method. Hence we will split the middle term such that the product of the two terms is the multiplication of the first term and the last term of the quadratic expression. Hence we will further simplify the expression by taking common terms together and hence we get the factor of the given expression.

Complete step by step solution:
Now we know that the given equation is a quadratic expression of the form $a{{x}^{2}}+bx+c$ .
Now to factorize the equation we will use splitting the middle term method.
In this method we will split the middle term such that the product of the terms obtained after splitting is equal to the product of first term and the last term of the expression.
Hence we will split 4x as 6x – 2x.
Now we have $\left( 6x \right)\left( -2x \right)=-12{{x}^{2}}=-12\times {{x}^{2}}$
Hence the given expression becomes $2{{x}^{2}}+6x-2x-6$
Now in the above expression we will group common terms, Hence taking 2x common from the first two terms and then taking -2 common from the last two terms we get \[\Rightarrow 2x\left( x+3 \right)+2\left( x+3 \right)\]
Now taking x + 3 common we get,
\[\Rightarrow \left( 2x+2 \right)\left( x+3 \right)\]
Hence the given expression is factored.

Note: Now note the value of which for which the expression is 0 is called the root of the polynomial. Now if we have the factorization of the expression as $\left( x-\alpha \right)\left( x-\beta \right)$ then we can see that on $x=\alpha ,\beta $ the value of expression is 0. Hence we can easily find the roots of the expression using factors.