Question

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

f(x) = 6x² – 3 – 7x

Answer 1

Given: f(x) = 6x² – 3 – 7x

Step-wise calculation:

1. Write in standard form:

f(x) = 6x² – 7x – 3

2. Factor by splitting the middle term:

6x² – 7x – 3 = 6x² – 9x + 2x – 3

= 3x(2x – 3) + 1(2x – 3)

= (2x – 3)(3x + 1)

3. Set f(x) = 0 and solve factors:

(2x – 3)(3x + 1) = 0

⇒ 2x – 3 = 0 or 3x + 1 = 0

⇒ `x = 3/2` or `x = -1/3`

4. Verify relationships for ax² + bx + c, a = 6, b = –7, c = –3: 

Sum of zeros = `(3/2) + (-1/3)`

= `9/6 - 2/6`

= `7/6`

= `-b/a`

= `-(-7)/6`

Product of zeros = `(3/2) xx (-1/3)` 

= `-1/2`

= `c/a`

= `(-3)/6`

The zeros are `x = 3/2` and `x = -1/3` and they satisfy sum = `-b/a` and product = `c/a`, as verified.