Solve each of the following quadratic equations:
x2 + 6x - (a2 + 2a - 8) = 0
x2 + 6x - (a2 + 2a - 8) = 0
Using the splitting middle term - the middle term of the general equation is divided in two such values that:
Product = a.c
For the given equation a = 1; b = 6 ;c = - (a2 + 2a - 8)
= 1. - (a2 + 2a - 8)
= - (a2 + 2a - 8)
And either of their sum or difference = b
= 6
Thus the two terms are (a + 4) and - (a - 2)
Difference = a + 4 - a + 2
= 6
Product = (a + 4) - (a - 2)
= - (a2 + 2a - 8)
⇒ x2 + 6x - (a2 + 2a - 8) = 0
⇒ x2 + (a + 4)x - (a - 2)x - (a + 4)(a - 2) = 0
⇒ x [x + (a + 4)] - (a - 2) [x + (a + 4)] = 0
⇒ [x + (a + 4)] [x - (a - 2)] = 0
⇒ [x + (a + 4)] = 0 or [x - (a - 2)] = 0
x = - (a + 4) or x = (a - 2)
Hence the roots of equation are - (a + 4) or (a - 2)