Quadratic Equation Level - 3

f(x) = ax² + bx + c & a < 0.
The equation f(x) = 0 has two distinct roots which is from the set {-2,-1, 0, 1, 2}. How many different pairs of roots of f(x) are possible such that f(0) is greater than or equals to 0?

px² + qx + r = 0 has one root greater than 3 and other root less than 1. Which of the following is necessarily true?

If f(x, y) = 4^xʸ + x^4ʸ then what is the total number of solutions for the equation f(x, y) = 5.

Directions for question number 4 & 5:

Which of the following equation has roots –c, –d.

The values of a quadratic function f(x) is a positive for all values of x, except for x = 4. If f(0) = 10. Find the value of f(–4).

Find all values of ‘a’, such that 4 lies somewhere between the roots of the equation 3x² + 4ax + (a + 3) = 0 for all values of x.

f(x) = x³ – (5 + k)x² + (6 + 5k)x – 6k, where ‘k’ is an odd prime number and k > 3. What is the range of values of x for which f(x) < 0.

f(x, p) = a(x – p)² + b(x – p) + c, where a, b, c are constants and a < 0 and ‘p’ is a natural number. It is given that the roots of the equation ax² + bx + c = 0 are 3, 4. Then, the value of x at which f(x, 5) attains its maximum value is:

f(x) = [x]² – 11[x] + 30, where [x] represents the largest integer less than or equal to x, then what is the sum of all integer solutions of the equation f(x) = 0