Q. 42

Solve each of the following quadratic equations:

x2 + 5x - (a2 + a - 6) = 0

Answer :

x2 + 5x - (a2 + 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 = - (a2 + a - 6)


= 1. - (a2 + a - 6)


= - (a2 + 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)]


= - (a2 + a - 6)


x2 + 5x - (a2 + 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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