Question

Verify the relationship between the zeroes and the coefficients of the quadratic polynomial 25x² – 16.

Answer 1

Given: Quadratic polynomial p(x) = 25x² – 16

Step-wise calculation:

1. Identify coefficients:

a = 25, b = 0, c = –16

2. Use the standard relations for a quadratic’s zeroes α, β:

`α + β = (-b)/a` and `αβ = c/a`, as stated in the reference

3. Compute sum:

`α + β = -b/a` 

= `-0/25`

= 0

4. Compute product:

`αβ = c/a` 

= `(-16)/25`

= `-16/25`

5. Solve directly:

25x² – 16 = 0

⇒ `x^2 = 16/25`

⇒ `x = ± 4/5`

So, the zeroes are `4/5` and `-4/5`.

6. Verify: sum = `4/5 + (-4/5) = 0`   ...(Matches `-b/a`)

Product = `(4/5)(-4/5) = -16/25`   ...(Matches `c/a`).

The relationship is verified: sum of zeroes = 0 = `-b/a` and product of zeroes = `-16/25 = c/a`. The zeroes are `4/5` and `-4/5`.