Mind Challenge:Answer the question:
True or False: All quadratic functions have x-intercepts. Why or why not?

Answer:False. Not all quadratic functions have x-intercepts. A quadratic function is typically represented in the standard form as:f(x) = ax² + bx + cThe x-intercepts of a quadratic function occur where the function equals zero (i.e., f(x) = 0). The number of x-intercepts is determined by the discriminant, which is part of the quadratic formula:x = -b ± (√b²-4ac) /2aThe discriminant ( D = b² - 4ac) can yield three scenarios: Two Real and Distinct Solutions: If D > 0, the quadratic function has two x-intercepts.One Real Solution: If D = 0, the quadratic function has one x-intercept (the vertex touches the x-axis).No Real Solutions: If D < 0, the quadratic function has no x-intercepts (the graph does not touch or cross the x-axis). Thus, a quadratic function can exist entirely above or below the x-axis, meaning it may not have any x-intercepts at all. For example, the function f(x) = x² + 1 has no x-intercepts because it is always above the x-axis.

Answered by alixzamarirapacon | 2024-10-17