Multiple choice

Directions: Select the correct alternative from the given choices. The graph of a quadratic expression, ax2 + bx + c, attains a maximum value of 10 at x = 2. If the graph intersects the x-axis at two points, one positive and the other negative, then which of a, b and c is/are positive?

  1. Only c.

  2. Only b.

  3. Both b and c.

  4. Both a and c.
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

A quadratic with a maximum value must have a < 0 (downward parabola). The vertex is at x = 2, so -b/(2a) = 2, meaning b = -4a. Since a < 0, b must be positive. The y-intercept is c. Since the parabola has a maximum of 10 at x=2 and crosses the x-axis at positive and negative values, the y-intercept (c) must be positive to allow the curve to pass through the positive y-axis region.

AI explanation

Since the quadratic attains a maximum, the coefficient of x2 is negative, meaning a is less than zero. The maximum occurs at x equals negative b divided by 2a, so 2 equals negative b divided by 2a, and since a is negative, b must be positive. The graph intersects the x-axis at one positive and one negative point, meaning the product of the roots is negative, which implies c divided by a is negative, making c positive. Therefore, both b and c are positive.

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