middle term splitting of 2x2-11x+12
Answers
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!