Answer:
(a) The zeroes of f(x) are rational numbers
(b) The smallest integer value of a is 3
Explanation:
In the quadratic function f(x) = ax² + bx + c, we can know the number and the types of its roots using the discriminant Δ, where Δ = b² - 4ac
- If Δ > 0, then the zeroes are two different real numbers
- If Δ = 0, then the zeroes are the same real number
- If Δ < 0, then the zeroes are imaginary numbers
Note: In the first case if Δ is a perfect square number then the zeroes are rational, if not then the zeroes are irrational
(a)
∵ The equation function is f(x) = ax² + 11x + 12
∵ a = 2
∴ a = 2, b = 11 and c = 12
→ Substitute them in the rule of Δ
∵ Δ = (11)² - 4(2)(12) = 121 - 96
∴ Δ = 25
∵ Δ > 0
∴ f(x) has two different real zeroes
∵ 25 is a perfect square number
→ That means the zeroes of the function are rational numbers
∴ The zeroes of f(x) are rational numbers
(b)
∵ f(x) ha imaginary zeroes
→ By using the third rule above
∴ Δ < 0
∵ a = a, b = 11 and c = 12
→ Substitute them in the rule of Δ
∴ (11)² - 4(a)(12) < 0
∴ 121 - 48a < 0
→ Add 48a to both sides
∵ 121 - 48a + 48a < 0 + 48a
∴ 121 < 48a
→ Divide both sides by 48
∴ 2.52083 < a
∴ a > 2.52083
∵ The smallest integer greater than 2.52083 is 3
∴ The smallest integer value of a is 3