Question

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

g(s) = 4s² – 4s + 1

Answer 1

Given: g(s) = 4s² – 4s + 1.

Step-wise calculation:

1. Use the quadratic formula

`s = (-b ± sqrt(b^2 - 4ac))/(2a)` with a = 4, b = –4, c = 1

2. Discriminant Δ = b² – 4ac

= (–4)² – 4 × 4 × 1 

= 16 – 16

= 0

3. `s = (-(-4) ± sqrt(0))/(2 xx 4)`

= `(4 ± 0)/8`

= `4/8`

= `1/2`

Both zeros are `s = 1/2` (a repeated root).

4. Sum of zeros = `1/2 + 1/2 = 1`.

Product of zeros = `1/2 xx 1/2 = 1/4`.

5. From coefficients: `-(b)/a = -(-4)/4 = 1` and `c/a = 1/4`.

The zeros are `s = 1/2` (double root). The sum and product of the zeros equal `(-b)/a` and `c/a` respectively, so the relationship between the zeros and the coefficients is verified.