Question

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

`h(s) = 2s^2 - (1 + 2sqrt(2))s + sqrt(2)`

Answer 1

Given: `h(s) = 2s^2 - (1 + 2sqrt(2))s + sqrt(2)`

Step-wise calculation:

1. Identify coefficients:

`a = 2, b = -(1 + 2sqrt(2)), c = sqrt(2)`

2. Discriminant: Δ = b² – 4ac 

`b^2 = (1 + 2sqrt(2))^2`

= `9 + 4sqrt(2)` 

So `Δ = (9 + 4sqrt(2)) - 4 xx 2 xx sqrt(2)` 

= `9 - 4sqrt(2)`

3. Simplify `sqrt(Δ)`:

`9 - 4sqrt(2) = (1 - 2sqrt(2))^2` 

So `sqrt(Δ) = 2sqrt(2) − 1` (positive root).

4. Roots by quadratic formula:

`s = (-b ± sqrt(Δ))/(2a)` 

= `(1 + 2sqrt(2) ± (2sqrt(2) - 1))/4`

`s_1 = (1 + 2sqrt(2) + 2sqrt(2) - 1)/4` 

= `(4sqrt(2))/4` 

= `sqrt(2)`

`s_2 = (1 + 2sqrt(2) - 2sqrt(2) + 1)/4` 

= `2/4` 

= `1/2`

5. Verify relationship between zeros and coefficients:

For a quadratic ax² + bx + c, sum of roots = `-b/a` and product = `c/a`.

Sum: `s_1 + s_2 = sqrt(2) + 1/2` 

= `(1 + 2sqrt(2))/2`

= `-b/a`

Product: `s_1 xx s_2 = sqrt(2) xx 1/2` 

= `sqrt(2)/2`

= `c/a`

The zeros of h(s) are `s = sqrt(2)` and `s = 1/2`. They satisfy sum = `(1 + 2sqrt(2))/2` and product = `sqrt(2)/2`, which match `-b/a` and `c/a` respectively, so the relationship between zeros and coefficients is verified.