Finding the Zeroes of the Polynomial
To find the zeroes of the polynomial 4√5x² - 17x + 3√5, we use the quadratic formula:
x = [ -b ± √(b² - 4ac) ] / 2a
Where:
- a = 4√5
- b = -17
- c = 3√5
Calculating the discriminant (D):
- D = b² - 4ac
- D = (-17)² - 4(4√5)(3√5)
- D = 289 - 240 = 49
Since the discriminant is positive, there are two real and distinct roots.
Finding the roots:
- x = [17 ± √49] / (2 * 4√5)
- x = [17 ± 7] / (8√5)
Calculating the two zeroes:
1. x₁ = (24) / (8√5) = 3/√5
2. x₂ = (10) / (8√5) = 5/4√5
Verification of Relation Between Zeroes and Coefficients
For any quadratic polynomial ax² + bx + c, the relationships are:
- Sum of zeroes (α + β) = -b/a
- Product of zeroes (αβ) = c/a
Using the found zeroes:
- Sum: α + β = 3/√5 + 5/4√5 = (12 + 5) / 4√5 = 17/4√5
- Product: αβ = (3/√5)(5/4√5) = 15/20 = 3/4
Calculating coefficients:
- -b/a = 17 / (4√5) (matches sum)
- c/a = (3√5) / (4√5) = 3/4 (matches product)
Conclusion
The zeroes of the polynomial 4√5x² - 17x + 3√5 are validated against the coefficients, confirming the relationships hold true. Thus, both the sum and product of the zeroes align with the corresponding coefficients of the polynomial.