Question

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

2x² – x – 6

Answer 1

Given: 2x² – x – 6. (Here a = 2, b = –1, c = –6.)

Step-wise calculation:

1. Solve 2x² – x – 6 = 0.

2. Factor by splitting the middle term:

2x² – x – 6 = 2x² – 4x + 3x – 6

= 2x(x – 2) + 3(x – 2)

= (2x + 3)(x – 2)

3. Set each factor = 0:

2x + 3 = 0 

⇒ `x = -3/2` 

x – 2 = 0 

⇒ x = 2

4. Sum and product of the zeros:

Sum = `(-3/2) + 2 = 1/2` 

Product = `(−3/2) × 2 = -3`

5. Compare with coefficients known relations:

Sum = `-b/a`

Product = `c/a`

`-b/a = -(-1)/2`

= `1/2` (matches sum)

`c/a = (-6)/2`

= –3 (matches product)

The zeros are `x = -3/2` and x = 2 and they satisfy the relations sum = `-b/a` and product = `c/a`, so the relationship between zeros and coefficients is verified.