Solve each of the following quadratic equations:

x^{2} - 2ax - (4b^{2} - a^{2}) = 0

x^{2} - 2ax - (4b^{2} - a^{2}) = 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 = - 2a c = - (4b^{2} - a^{2})

= 1. - (4b^{2} - a^{2})

= - (4b^{2} - a^{2})

And either of their sum or difference = b

= - 2a

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

Difference = 2b - a - 2b - a

= - 2a

Product = (2b - a) - (2b + a)

(∵ using a^{2} - b^{2} = (a + b) (a - b))

= - (4b^{2} - a^{2})

x^{2} - 2ax - (4b^{2} - a^{2}) = 0

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

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

⇒ [x + (2b - a)] [x - (2b + a)] = 0

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

⇒ x = (a - 2b) or x = (a + 2b)

Hence the roots of given equation are (a - 2b) or x = (a + 2b)

43