Question

Find the zeros of the following quadratic polynomial and verify the relationship between the zeros and the coefficients:

`f(x) = x^2 - (sqrt(3) + 1)x + sqrt(3)`

Answer 1

Given: `f(x) = x^2 - (sqrt(3) + 1)x + sqrt(3)`

Step-wise calculation:

1. Try factoring: `(x - 1)(x - sqrt(3)) = x^2 - (1 + sqrt(3))x + sqrt(3)`, which equals f(x).

2. So `f(x) = (x - 1)(x - sqrt(3))`.

Hence zeros are x = 1 and x = `sqrt(3)`.

3. For verification, recall for ax² + bx + c the sum of zeros = `-b/a` and product = `c/a`.

Here `a = 1, b = -(sqrt(3) + 1), c = sqrt(3)`.

Sum of zeros = `1 + sqrt(3)`

= `sqrt(3) + 1`

= `-b/a`

Product of zeros = `1 xx sqrt(3)`

= `sqrt(3)`

= `c/a`

The zeros are x = 1 and x = `sqrt(3)`, and they satisfy sum = `sqrt(3) + 1 = -b/a` and product = `sqrt(3) = c/a`, so the relationship between zeros and coefficients is verified.