Question

How do you factor $ 3{x^2} = 16x + 12 $ ?

Answer 1

Hint: Move all the terms on one side of the equation and then find the factors by using the split method I.e. splitting the middle term and then find the factors for the values for the “x”.

Complete step by step solution:
Take the given expression: $ 3{x^2} = 16x + 12 $
Move all the terms on one side of the equation. When you move any term from one side to another then the sign of the term also changes. Positive terms become negative and vice-versa.
 $ 3{x^2} - 16x - 12 = 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 ( - 12) = - 36 $
Now, you have to split the middle term to get $ - 36 $ in multiplication and addition or subtraction to get the middle term i.e. $ - 16 $ . 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.
 $
   - 36 = ( - 18) \times (2) \\
  ( - 16) = ( - 18) + 2 \;
  $ $ $ $ $
Write the equivalent value for the middle term –
 $ 3{x^2} - 18x + 2x - 12 = 0 $
Now, make the pair of two terms in the above equation-
 $ \underline {3{x^2} - 18x} + \underline {2x - 12} = 0 $
Find the common factors from the paired terms –
 $ 3x(x - 6) + 2(x - 6) = 0 $
Take the common factors in the above equation –
 $ (x - 6)(3x + 2) = 0 $
 $
   \Rightarrow x - 6 = 0 \\
   \Rightarrow x = 6 \;
  $
or
 $
   \Rightarrow 3x + 2 = 0 \\
   \Rightarrow x = - \dfrac{2}{3} \;
  $
Hence, factors are $ x = 6, - \dfrac{2}{3} $
So, the correct answer is “ $ x = 6, - \dfrac{2}{3} $ ”.

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.