Question

If a and b are the zeros of the quadratic polynomial x^2-x-2 find a polynomial whose zeros are 2a+1 and 2b+1

Answer 1

(x+1)(x-5) = x^2-4x-5

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Answer 2

Given: a and b are the zeros of given polynomial x^2 - x - 2

Factorizing the given quadratic polynomial by the middle term splitting method.

$=x^{2} -x -2\\= x^{2} -(2-1)x-2\\ =x^{2} -2x+1x-2\\ =x(x-2)+1(x-2) \\= (x-2)(x+1)$

The zeros of quadratic polynomial are x = 2 and x = - 1

According to the problem,

a = 2 and b = - 1 or a = - 1 and b = 2

Now, we shall find the quadratic equation whose zeros are (2 a + 1) and (2 b + 1).

Case - 1:

If, a = 2 and b = - 1 then (2 a + 1) = 5 and (2 b + 1 ) = - 1

Hence, the required quadratic polynomial will be x^2 - (5 - 1) x - 5

Or, $(x^{2} -4x-5)$

Case - 2:

If, a = - 1 and b = 2 then (2 a + 1) = - 1 and (2 b + 1 ) = 5

Hence, the required quadratic polynomial will be x^2 - ( - 1 + 5) x - 5

Or, $(x^{2} -4x-5)$

Hence, in both the cases, the required quadratic polynomial will $(x^{2} -4x-5)$