Obtain a quadratic polynomial with following conditions?? 1. the sum of zeroes=1/3,the product of zeroes=1/2
2 Answers
Explanation:
The quadratic formula is
Sum of two roots:
Product of two roots:
We have
Proof:
# 6x^2 - 2x + 3 = 0#
Explanation:
If we have a general quadratic equation:
# ax^2 + bx + c = 0 iff x^2 + b/ax + c/a = 0#
And we denote the root of the equation by
# (x-alpha)(x-beta) = 0 iff x^2 - (alpha+beta)x + alpha beta = 0 #
Which gives us the well studied properties:
# {: ("sum of roots", = alpha+beta, = -b/a), ("product of roots", = alpha beta, = c/a) :} #
Thus we have:
# {: ( alpha+beta, = -b/a, = 1/3), ( alpha beta, = c/a, =1/2) :} #
So the sought equation is:
# x^2 - "(sum of roots)"x + "(product of roots)" = 0#
i.e.:
# x^2 - 1/3x + 1/2 = 0#
And (optionally), to remove the fractional coefficients, we multiply by
# 6x^2 - 2x + 3 = 0#