Question

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

`phi(x) = 4x^2 + 5sqrt(2)x - 3`

Answer 1

1. Finding the zeros

Set the polynomial to zero and split the middle term:

`4x^2 + 6sqrt(2)x - sqrt(2)x - 3 = 0`

`2sqrt(2)x(sqrt(2)x + 3) - 1(sqrt(2)x + 3) = 0`

`(sqrt(2)x + 3)(2sqrt(2)x - 1) = 0`

`sqrt(2)x + 3 = 0 ⇒ x = -3/sqrt(2)`

`2sqrt(2)x - 1 = 0 ⇒ x = 1/(2sqrt(2))`

2. Verification of relationships

For `phi(x) = 4x^2 + 5sqrt(2)x - 3`, the coefficient are `a = 4, b = 5sqrt(2)` and c = –3

Sum of zeros:

`α + β = 1/(2sqrt(2)) - 3/sqrt(2)`

= `(1 - 6)/(2sqrt(2))`

= `-5/(2sqrt(2))`

= `-(5sqrt(2))/4`

`-b/a = -(5sqrt(2))/4`

LHS = RHS (Verified)

Product of zeros:

`α xx β = (1/(2sqrt(2))) xx (-3/(sqrt(2)))`

= `-3/4`

`c/a = -3/4`

LHS = RHS (Verified)