Question

middle term splitting of 2x2-11x+12​

Answer 1

p(x) = 2x² - 11x + 12 = 0

Here we have to split the middle term  - 11x  as sum of two variables.

Let the coefficient of x, -11, be split as p + q.  Let  -11 = p + q.

Actually p and q are in the form  u + v  and  u - v respectively, or vice versa.

Let  p = u + v  and  q = u - v.

So,

    p + q = - 11

⇒  u + v + u - v = - 11

⇒  2u = - 11

⇒  u = - 11 / 2

Okay, 'u' is got.

And also,  pq  values the product of coefficient of x² and constant term.

    pq = 2 · 12

⇒  (u + v)(u - v) = 24

⇒  u² - v² = 24

⇒  (- 11 / 2)² - v² = 24

⇒  (121 / 4) - v² = 24

⇒  v² = (121 / 4) - 24

⇒  v² = (121 - 96) / 4

⇒  v² = 25 / 4

⇒  v = 5 / 2

Okay, 'v' too is got.

Now,

p = u + v = (- 11 / 2) + (5 / 2) = (- 11 + 5) / 2 = - 6 / 2 = - 3

q = u - v = (- 11 / 2) - (5 / 2) = (- 11 - 5) / 2 = - 16 / 2 = - 8

Thus, -11 should be split as (-3) + (-8).

Hence,

11x  =  - 3x - 8x =  - 8x - 3x

Hence split!

Now,

    2x² - 11x + 12

⇒  2x² - 8x - 3x + 12

⇒  2x(x - 4) - 3(x - 4)

⇒  (x - 4)(2x - 3)

Hence factorized!