Question

Let f(x) = ax² + bx + c, where a, b, and c are real numbers with a ≠ 0. If the graph of f(x) intersects the x-axis at two distinct points with x-coordinates p and q (p < q), then the equation ax² + bx + c + 1 = 0 has:

• A. Exactly one root between p and q
• B. No roots between p and q
• C. Two roots, both between p and q
• D. One root less than p and one root greater than q

Answer 1

Answer: (B)

No roots between p and q

Answer 2

Since f(x) intersects the x-axis at p and q, these are the roots of the equation ax² + bx + c = 0.

Now, consider the equation ax² + bx + c + 1 = 0. This equation is obtained by shifting the graph of f(x) upwards by 1 unit.

When we shift the graph of a quadratic function upwards, the roots move closer to each other. Since the original roots (p and q) were on the x-axis, shifting the graph upwards by 1 unit will place both new roots above the x-axis.

Therefore, the equation ax² + bx + c + 1 = 0 has no roots between p and q.

So, the correct answer is (B) No roots between p and q.