Question Video: Identifying the x-axis Intersections of a Quadratic Function | Nagwa Question Video: Identifying the x-axis Intersections of a Quadratic Function | Nagwa

Question Video: Identifying the x-axis Intersections of a Quadratic Function Mathematics • Third Year of Preparatory School

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If 𝑓(π‘₯) = π‘₯Β² βˆ’ π‘₯ βˆ’ 6, and the solutions to the equation π‘₯Β² βˆ’ π‘₯ βˆ’ 6 = 0 are {3, βˆ’2}, then what are the points where 𝑓(π‘₯) intercepts the π‘₯-axis?

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Video Transcript

If 𝑓 of π‘₯ is equal to π‘₯ squared minus π‘₯ minus six and the solutions to the equation π‘₯ squared minus π‘₯ minus six equals zero are three and negative two, then what are the points where 𝑓 of π‘₯ intercepts the π‘₯-axis?

In this question, we have been given a quadratic function 𝑓 of π‘₯ equals π‘₯ squared minus π‘₯ minus six. We have also been told that the equation π‘₯ squared minus π‘₯ minus six equals zero has two solutions, which are three and negative two.

We can recall that the solution set of a quadratic function can be found by setting 𝑓 of π‘₯ equal to zero. In our case, we have that when 𝑓 of π‘₯ is equal to zero, π‘₯ squared minus π‘₯ minus six is also equal to zero. Hence, we can say that the solutions to 𝑓 of π‘₯ are three and negative two. Finding this is very useful in helping us solve the question, since we know that the points where 𝑓 of π‘₯ will intercept the π‘₯-axis will be when 𝑓 of π‘₯ is equal to zero.

Let’s take a minute to think as to why this is the case. The equation we are sketching the graph of is 𝑦 equals 𝑓 of π‘₯. The π‘₯-axis is the same as the equation 𝑦 equals zero. Hence, the points where 𝑓 of π‘₯ intercepts the π‘₯-axis will be when these two lines intersect. So that is when 𝑓 of π‘₯ is equal to zero.

We have already found that when 𝑓 of π‘₯ equals zero, π‘₯ is equal to three or negative two. Hence, these are the π‘₯-coordinates of the points where 𝑓 of π‘₯ intercepts the π‘₯-axis. Since both of these points are on the π‘₯-axis, we know that their 𝑦-coordinates must be zero.

Therefore, our solution to this question is that the points where 𝑓 of π‘₯ intercepts the π‘₯-axis are three, zero and negative two, zero. Here is a quick sketch of what a graph of 𝑓 of π‘₯ may look like. We can see that it intersect the π‘₯-axis at three, zero and negative two, zero.

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