2012 DSE Maths MC Q34

The answer : (C) The roots of the equation of f(x) = 4 are real numbers

The vertex of the graph of y = f(x) is (3, -4).
Then, y = f(x) = a(x - 3)² - 4

A. false
f(x) = 0
a(x - 3)² - 4 = 0
a(x - 3)² = 4
(x - 3)² = 4/a
x - 3 = ±√(4/a)
x = 3 ± √(4/a)
The roots are NOT integers when √(4/a) is NOT an integer.

B. false
f(x) - 3 = 0
a(x - 3)² - 4 - 3 = 0
a(x - 3)² = 7
(x - 3)² = 7/a
x - 3 = ±√(7/a)
x = 3 ± √(7/a)
The roots are NOT rational numbers when √(7/a) is NOT rational number.

C. true
f(x) + 4 = 0
a(x - 3)² - 4 + 4 = 0
a(x - 3)² = 0
(x - 3)² = 0
x - 3 = 0
x = 3 (double real roots)

D. false
f(x) + 5 = 0
a(x - 3)² - 4 + 5 = 0
a(x - 3)² = -1
(x - 3)² = -1/a
x - 3 = ±√(-1/a)
x = 3 ± √(-1/a)
The roots are real numbers when a < 0.