Solve each of the following quadratic equations:

x^{2} + 5x - (a^{2} + a - 6) = 0

x^{2} + 5x - (a^{2} + a - 6) = 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 = 5; c = - (a^{2} + a - 6)

= 1. - (a^{2} + a - 6)

= - (a^{2} + a - 6)

And either of their sum or difference = b

= 5

Thus the two terms are (a + 3) and - (a - 2)

Difference = a + 3 –a + 2

= 5

Product = (a + 3). - (a - 2)

= - [(a + 3)(a - 2)]

= - (a^{2} + a - 6)

x^{2} + 5x - (a^{2} + a - 6) = 0

⇒ x ^{2} + (a + 3)x - (a - 2)x - (a + 3)(a - 2) = 0

⇒ x[x + (a + 3)] - (a - 2) [x + (a + 3)] = 0

⇒ [x + (a + 3)] [x - (a - 2)] = 0

⇒ [x + (a + 3)] = 0 or [x - (a - 2)] = 0

⇒ x = - (a + 3) or x = (a - 2)

Hence the roots of given equation are - (a + 3) or (a - 2)

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