### Video Transcript

If the equation π of π₯ is equal to zero has two solutions, does this mean that π
is a quadratic function that intercepts the π₯-axis at two points?

To answer this question, we first note that if the equation π of π₯ is equal to zero
has two solutions π sub one and π sub two, then its solution set π has two
elements. And this is where π sub one and π sub two are distinct real numbers. We also know that for a quadratic function of the form π of π₯ is equal to ππ₯
squared plus ππ₯ plus π, if the equation π of π₯ is equal to zero has two
solutions π₯ is equal to π sub one and π₯ is π sub two, then π sub one and π sub
two are two distinct roots of the function π of π₯. And letβs consider what this means graphically.

We know that the graph of a quadratic function is a parabola, that is, a curve thatβs
either U shaped or n shaped. And itβs symmetric about a vertical line. The sign of the leading coefficient π β thatβs the coefficient of π₯ squared β tells
us whether the curve is going to be U shaped or n shaped. If π is less than zero, the curve is n shaped as shown. If, on the other hand, π is greater than zero as shown in the second graph, the
curve is U shaped.

Recall also that the number of solutions to the equation π of π₯ is equal to zero
tells us whether and if so at how many points the curve intercepts the π₯-axis. If there are two distinct solutions π sub one and π sub two, then the function has
two distinct roots and the curve intercepts and in this case crosses the π₯-axis
twice. If thereβs one repeated solution so that π of π₯ is equal to π multiplied by π₯
minus π sub one squared, where π is nonzero, then the graph intercepts the π₯-axis
only once, touching it at π₯ is equal to π sub one. In this case, we say that π sub one is a repeated root.

The solution set π then has only one element and thatβs π sub one. And if there are no real solutions to π of π₯ is equal to zero, then the graph does
not intercept the π₯-axis and we say that the solution set is the empty set.

Okay, so now making some space, weβve established in fact that if π of π₯ is a
quadratic function whose graph intercepts the π₯-axis at two points, then the
equation π of π₯ is equal to zero has two real solutions. But this does not quite match the given statement in the question. In fact, the question statement is the reverse of this. Letβs read this again. This says that if π of π₯ is equal to zero has two solutions, does this mean that π
is a quadratic function that intersects the π₯-axis at two points? In fact, it turns out that there are other functions π of π₯ that are not
quadratics, but for which there are exactly two solutions to π of π₯ is equal to
zero.

We can see some examples of these as shown. We know that a quadratic has exactly one turning point. However, we see that while each of the graphs of the functions shown intercept the
π₯-axis at two points, they each have more than one turning point and so they cannot
be quadratic functions. And hence, the fact that the equation π of π₯ is equal to zero has two solutions for
a function π of π₯ does not necessarily mean that π of π₯ is a quadratic
function. Our answer is therefore no, π of π₯ may not necessarily be a quadratic function.