Question

For the quadratic equation (y - c) (y - d) + 2 = 0 to have real roots, which of the following could be true?

1. (c - d) > (8)^3/2 and (c - d) ≤ (-8) ^(3/2)
2. (c - d) > (2) and (c - d) ≤ (-2)
3. (c - d) > (2)^3/2 and (c - d) ≤ (-2)^(3/2)
4. (c + d) > (2)^3/2 and (c + d) ≤ (-2)^(3/2)

Answer 1

Correct Answer - Option 3 : (c - d) > (2)^3/2 and (c - d) ≤ (-2)^(3/2)

Calculation:

The given equation can be written as y² - (d + c) y + cd + 2 = 0 

For real roots, D = (d + c)² - 4(cd + 2) > 0

Or d² + c² - 2cd - 8 > 0

Or (c - d)² > 8

Or (c - d) > (2)^3/2 and (c-d) ≤ (-2)^(3/2)