Final answer:
If the graph of a parabola does not cross the x-axis, it implies that the quadratic equation representing the parabola has complex roots, because the discriminant (b² - 4ac) is less than 0.the correct answer to the question is: d) It has complex roots.
Step-by-step explanation:
If the graph of a parabola does not cross the x-axis, it means that the quadratic equation representing the parabola has no real roots. Such a parabola will either lie entirely above the x-axis or entirely below it, depending on whether the quadratic function opens upwards or downwards, respectively. To understand this, consider the general form of a quadratic equation ax²+bx+c = 0.
The roots of a quadratic equation are the solutions to this equation, which can be found using the quadratic formula:
x = ∛( -b ± √(b² - 4ac) ) / (2a)
The part b² - 4ac under the square root in the quadratic formula is called the discriminant. The discriminant determines the nature of the roots:
- If b² - 4ac > 0, the equation has two distinct real roots.
- If b² - 4ac = 0, the equation has exactly one real root, which means the parabola touches the x-axis at a single point (vertex).
- If b² - 4ac < 0, the equation has no real roots, meaning the graph of the parabola does not cross the x-axis at all, and instead, the roots are complex.
Since the parabola in question does not cross the x-axis, it implies that b² - 4ac < 0. Therefore, the equation has two complex roots. Consequently, the correct answer to the question is: d) It has complex roots.