Question

Verify the relation between the zeroes and the coefficients of the quadratic polynomial 4x² – 9.

Answer 1

1. Identify coefficients

Compare the given polynomial 4x² – 9 with the standard quadratic form ax² + bx + c:

a = 4   ...(Coefficient of x²)

b = 0   ...(Coefficient of x, since the term is missing)

c = –9   ...(Constant term)

2. Find the zeroes

To find the zeroes, set the polynomial equal to zero:

4x² – 9 = 0

Using the difference of squares identity a² – b² = (a – b)(a + b):

(2x)² – (3)² = 0

(2x – 3)(2x + 3) = 0

This gives:

2x – 3 = 0

⇒ `x = 3/2`

2x + 3 = 0

⇒ `x = -3/2`

Thus, the zeroes are `α = 3/2` and `β = -3/2`.

3. Verify sum of zeroes

The formula for the sum of zeroes is `α + β = - b/a`.

Sum of calculated zeroes: `3/2 + (-3/2) = 0`

Using coefficients: `- b/a = - 0/4 = 0`

Since 0 = 0, the first relation is verified.

4. Verify product of zeroes

The formula for the product of zeroes is `α xx β = c/a`.

Product of calculated zeroes: `(3/2) xx (-3/2) = - 9/4`

Using coefficients: `c/a = (-9)/4 = -9/4`

Since ` -9/4 = -9/4`, the second relation is verified.