Question

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

Answer 1

Hint : In this question, we need to solve the equation $2{x^2} - 3x - 2$ . For splitting the middle term into two factors, we will determine the factors that multiply to give $ac$ i.e., $2 \times - 2 = - 4$ , and add to give $b$ i.e., $- 3$ which is called sum-product pattern. Then, factor the first two and last two terms separately. If we have done this correctly, then two new terms will have a clearly visible common factor. Finally, we will equate the factors to $0$ and determine the value of $x$ .

Complete step-by-step answer :
Now, we need to solve $2{x^2} - 3x - 2$ .
First, let us determine the factors of the given equation.
According to the rule to factorize,
Product= ${x^2}$ coefficient $\times$ constant
And, sum= $x$ coefficient
Thus, we will find two numbers that multiply to give $ac$ i.e., $2 \times - 2 = - 4$ and add to give $b$ i.e., $- 3$ ,
Here, the product is negative. So, we can say that one of the factors is negative, and then the other is positive.
Now, let’s consider the possible factors and their sum.
$4 \times - 1 = - 4;4 + \left( { - 1} \right) = 3 \\\ 2 \times - 2 = - 4;2 + \left( { - 2} \right) = 0 \\\ 1 \times - 4 = - 4;1 + \left( { - 4} \right) = - 3 \;$
From this it is clear that the factors are $1$ and $- 4$ .
Now, by rewriting the middle term with those factors, we have,
$2{x^2} - 3x - 2 = 0$
$\left( {2{x^2} + x} \right) - \left( {4x + 2} \right) = 0$
Factor out the greatest common factor from each group,
$x\left( {2x + 1} \right) - 2\left( {2x + 1} \right) = 0$
Factor the polynomial by factoring out the greatest common factor, $2x + 1$ ,
$\Rightarrow \left( {x - 2} \right)\left( {2x + 1} \right) = 0$
Hence, the factors are $\left( {x - 2} \right)$ and $\left( {2x + 1} \right)$ .
So, the correct answer is “ $\left( {x - 2} \right)$ and $\left( {2x + 1} \right)$ ”.

Note : In this question it is important to note that this factorization method works for all quadratic equations. The standard form of the quadratic equation is $a{x^2} + bx + c = 0$ . It is called factoring because we find the factors. A factor is something we multiply by. There is no simple method of factoring a quadratic expression, but with a little practice it becomes easier. If the question is to solve the equation, then we can finally, equate the equation to $0$ which is common in all quadratic equations because we need to determine the value of the given unknown variable.