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.