Question

How do you solve $2{x^2} - 7x + 6 = 0$ by factoring?

Answer 1

Hint:Use sum product method to factorize the given equation and after finding the factors, compare the factors separately with zero in order to get the solution for the equation.In sum product method, the middle term (or the term which is coefficient of “x”) is being split into two parts in such a way that their product should equals to the product of first term (or the term which is coefficient of ${x^2}$) and the constant.

Complete step by step answer:
To solve the given equation $2{x^2} - 7x + 6 = 0$ with the help of factorization, we will use the sum product method to factorize the given equation.In this method we will first split the coefficient of $x$ such that their product should equals to the product of coefficient of ${x^2}$ and the constant. We need to find the factors of the product $2 \times 6 = 12$, so that we can find suitable numbers for splitting the middle term
$12 = 2 \times 2 \times 3$
We can see that the mid-term $7$ can be split into $2 \times 2 = 4\;{\text{and}}\;3$
So splitting the mid-term, we will get
$
\Rightarrow 2{x^2} - 7x + 6 = 0 \\
\Rightarrow 2{x^2} - (4x + 3x) + 6 = 0 \\
\Rightarrow 2{x^2} - 4x - 3x + 6 = 0 \\ $
Now taking the common factors out from first two and next two terms,
$
\Rightarrow 2{x^2} - 4x - 3x + 6 = 0 \\
\Rightarrow 2x(x - 2) - 3(x - 2) = 0 \\
\Rightarrow (x - 2)(2x - 3) = 0 \\ $
We get the factors \[(x - 2)\;{\text{and}}\;(2x - 3)\]
Comparing them to $0$ separately to get the solution
$
\Rightarrow x - 2 = 0\;{\text{and}}\;2x - 3 = 0 \\
\Rightarrow x = 2\;{\text{and}}\;2x = 3 \\
\Rightarrow x = 2\;{\text{and}}\;x = \dfrac{3}{2} \\ $
$\therefore x = 2\;{\text{and}}\;x = \dfrac{3}{2}$ is the required solution for the equation $2{x^2} - 7x + 6 = 0$.

Note: If you face problems in splitting the middle term of the expression, then first find for common factors in the equation among all the terms, it will simplify the digits and make it easier for you to find the factors through the sum product method.