Question

Let $ x = 2 $ be a root of $ y = 4{x^2} - 14x + q = 0. $ Then y is equal to
A. $ (x - 2)(4x - 6) $
B. $ (x - 2)(4x + 6) $
C. $ (x - 2)( - 4x - 6) $
D. $ (x - 2)( - 4x + 6) $

Answer 1

Hint: First of all place the value of “x” and get the value of the unknown term “q” and then placing the value of “q” in the equation find the factors by using the concept of split of the middle terms.

Complete step-by-step answer:
Take the given expression: $ y = 4{x^2} - 14x + q = 0. $
Place the given value for $ x = 2 $ in the above equation
 $ 4{(2)^2} - 14(2) + q = 0. $
Simplify the above equation –
 $ 16 - 28 + q = 0. $
Make the required term the subject and move all the constants on the opposite side. When you move any term from one side to another then the sign of the terms also changes. Positive term is changed to negative and negative is changed to positive.
 $ q = 28 - 16 $
Simplify
 $ q = 12 $
Now, place the above value in the given expression –
 $ 4{x^2} - 14x + 12 = 0. $
Split the middle term in such a way that its product is equal to the product of the first and the last term.
 $ 4{x^2} - 6x - 8x + 12 = 0. $
Make the pair of first two and the last two terms in the above expression.
 $ \underline {4{x^2} - 6x} - \underline {8x + 12} = 0. $
Find the common factor from both the groups
 $ 2x(2x - 3) - 4(2x - 3) = 0 $
Write the common factors-
 $ (2x - 3)(2x - 4) = 0 $
The common factor in the above expression is
 $
  2(2x - 3)(x - 2) = 0 \\
  (4x - 6)(x - 2) = 0 \;
  $
Hence, from the given multiple choices the option A is the correct answer.
So, the correct answer is “Option A”.

Note: Be careful about the sign convention while the split of the middle term. The product of two negative terms gives the resultant value in positive and sum of two negative terms gives value in negative. Be good in multiples and addition to get the middle term split.