Question

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

`p(x) = x^2 + 2sqrt(2)x - 6`

Answer 1

Given: `p(x) = x^2 + 2sqrt(2)x - 6`.

Step-wise calculation:

1. Identify coefficients:

a = 1, b = `2sqrt(2)`, c = –6

2. Discriminant: Δ = b² – 4ac 

= `(2sqrt(2))^2 - 4(1)(-6)` 

= 8 + 24

= 32

3. Square root of discriminant:

`sqrt(Δ) = sqrt(32)`

= `4sqrt(2)`

4. Quadratic formula:

`x = (-b ± sqrt(Δ))/(2a)` 

= `(-2sqrt(2) ± 4sqrt(2))/2`

First root: `x_1 = (-2sqrt(2) + 4sqrt(2))/2`

= `(2sqrt(2))/2`

= `sqrt(2)`

Second root: `x_2 = (-2sqrt(2) - 4sqrt(2))/2`

= `(-6sqrt(2))/2`

= `-3sqrt(2)`

5. So the zeros are `x = sqrt(2)` and `x = -3sqrt(2)`.

6. Verify relationship between zeros and coefficients (sum and product for ax² + bx + c): sum = `-b/a`, product = `c/a`.

Sum of zeros: `sqrt(2) + (-3sqrt(2))`

= `-2sqrt(2)`

= `-b/a` (since `b = 2sqrt(2))`

Product of zeros: `sqrt(2) xx (-3sqrt(2))`

= –3 × 2 

= –6 

= `c/a`

The zeros of p(x) are `sqrt(2)` and `-3sqrt(2)`. They satisfy sum `(α + β) = -b/a = -2sqrt(2)` and product `(αβ) = c/a = -6`, so the relationship between the zeros and the coefficients is verified.