Question

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

f(x) = 4x² + 4x – 3

Answer 1

Given: f(x) = 4x² + 4x – 3

Step-wise calculation:

1. Set f(x) = 0:

4x² + 4x – 3 = 0

2. Use the quadratic formula

`x = (-b ± sqrt(b^2 - 4ac))/(2a)` with a = 4, b = 4, c = –3: 

b² – 4ac = 4² – 4 × 4 × (–3) 

= 16 + 48

= 64

So sqrt = 8. 

`x = (-4 ± 8)/8`. 

Thus `x_1 = (-4 + 8)/8`

= `4/8`

= `1/2` 

And `x_2 = (-4 - 8)/8`

= `(-12)/8`

= `(-3)/2`

3. Sum of zeros: `α + β = 1/2 + (-3/2) = -1`. 

Product of zeros: `αβ = (1/2) xx (-3/2) = -3/4`.

4. Compare with coefficients: `-b/a = -4/4 = -1` and `c/a = (-3)/4 = −3/4`. 

The general relations sum = `-b/a` and product = `c/a` hold.

The zeros are `x = 1/2` and `x = -3/2`. The sum and product of the zeros equal `-b/a` and `c/a` respectively, so the relationship between zeros and coefficients is verified.