Question

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

`f(v) = v^2 + 4sqrt(3)v - 15`

Answer 1

Given: `f(v) = v^2 + 4sqrt(3)v - 15`

Step-wise calculation:

1. Set f(v) = 0:

`v^2 + 4sqrt(3)v - 15 = 0`

2. Split the middle term and factor:

`v^2 + 5sqrt(3)v - sqrt(3)v - 15 = (v + 5sqrt(3))(v - sqrt(3))`

Hence `f(v) = (v + 5sqrt(3))(v - sqrt(3))`.

3. Solve each factor = 0:

`v + 5sqrt(3) = 0`

⇒ `v = -5sqrt(3)`

`v - sqrt(3) = 0`

⇒ `v = sqrt(3)`

4. Verify relationships for ax² + bx + c with a = 1, b = `4sqrt(3)`, c = –15: 

Sum of zeros = `(-5sqrt(3)) + (sqrt(3))`

= `-4sqrt(3)`

= `-b/a`

= `-(4sqrt(3))/1`

Product of zeros = `(-5sqrt(3))(sqrt(3))`

= –5 × 3 

= –15 

= `c/a`

= `(-15)/1`

Zeros are `v = -5sqrt(3)` and `v = sqrt(3)`. The sum and product of these zeros match `-b/a` and `c/a` respectively, so the relationship between zeros and coefficients is verified.