Question

How do you solve $ 3{x^2} - 6x - 24 = 0 $ by factoring?

Answer 1

Hint: First of all we will find the common factors if any from all the three terms and then will use the method of splitting the middle term to get the factors of the given quadratic equation. We have to split the middle term in such a way that the product of the term is equal to the product of the first and last term.

Complete step by step solution:
Take the given expression: $ 3{x^2} - 6x - 24 = 0 $
Take the common factor common from the given expression: $ 3{x^2} - 6x - 24 = 0 $
Now we will use the concept of to split the middle term.
Here we have three terms in the given expression.
Now, multiply the constant in the first term with the last term.
i.e. $ 3 \times ( - 24) = ( - 72) $
Now, you have to split the middle term to get $ ( - 72) $ in multiplication and addition or subtraction to get the middle term i.e. $ ( - 6) $ . Here applying the basic concept of the product of two negative terms gives us the positive term and addition of two negative terms gives the value in the negative sign.
 $
   - 72 = ( - 12) \times (6) \\
  ( - 24) = ( - 12) + 6 \;
 $
Write the equivalent value for the middle term –
 $ 3{x^2} - 12x + 6x - 24 = 0 $
Now, make the pair of two terms in the above equation-
 $ \underline {3{x^2} - 12x} + \underline {6x - 24} = 0 $
Find the common factors from the paired terms –
 $ 3x(x - 4) + 6(x - 4) = 0 $
Take the common factors in the above equation –
 $ (x - 4)(3x + 6) = 0 $
Therefore, $ x = 4, - 2 $ and $ x = - 2 $
So, the correct answer is “ $ x = 4, $ and $ x = - 2 $ ”.

Note: Here we were able to split the middle term and find the factors but in case it is not possible then we can find factors by using the formula\[x = \dfrac{{ - b \pm \sqrt \Delta }}{{2a}}\] and considering the general form of the quadratic equation $ a{x^2} + bx + c = 0 $ . Be careful about the sign convention and simplification of the terms in the equation.