Question Video: Deciding Whether a Particular Graphical Property Implies That a Function Is a Quadratic | Nagwa Question Video: Deciding Whether a Particular Graphical Property Implies That a Function Is a Quadratic | Nagwa

# Question Video: Deciding Whether a Particular Graphical Property Implies That a Function Is a Quadratic Mathematics • Third Year of Preparatory School

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If the equation π(π₯) = 0 has two solutions, does this mean that π is a quadratic function that intercepts the π₯-axis at two points?

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### 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 ππ₯ squared plus ππ₯ 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 letβs consider what this means graphically.

We know that the graph of a quadratic function is a parabola, that is, a curve thatβs either U shaped or n shaped. And itβs symmetric about a vertical line. The sign of the leading coefficient π β thatβs 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 thereβs 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 thatβs 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, weβve 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. Letβs 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.

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