tiffany is solving an equation where both sides are quadratic expressions. she sets each quadratic equation…
tiffany is solving an equation where both sides are quadratic expressions. she sets each quadratic equation equal to y and graphs the resulting system. if the graph of one quadratic opens upward and the other opens downward, what is the greatest possible number of intersections for these graphs?\nnone\none\ntwo\ninfinitely many
Answer
Explanation:
Step1: Define quadratic equations
Let the upward-opening quadratic be $y = ax^2 + bx + c$ where $a>0$, and the downward-opening quadratic be $y = dx^2 + ex + f$ where $d<0$.
Step2: Find intersection condition
Set equations equal: $ax^2 + bx + c = dx^2 + ex + f$ Rearrange to: $(a-d)x^2 + (b-e)x + (c-f) = 0$
Step3: Analyze resulting equation
Since $a>0$ and $d<0$, $a-d = a + |d| > 0$, so this is a quadratic equation. A quadratic equation can have at most 2 distinct real solutions, which correspond to 2 intersection points.
Answer:
two