Question Video: Finding the Solution Set of a Quadratic Equation Graphically Mathematics

The graph shows the function 𝑓(π‘₯) = π‘₯Β² βˆ’ 2π‘₯ + 3. What is the solution set of 𝑓(π‘₯) = 0?

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Video Transcript

The graph shows the function 𝑓 of π‘₯ equals π‘₯ squared minus two π‘₯ plus three. What is the solution set of 𝑓 of π‘₯ equals zero.

Given some graph of a function 𝑦 equals 𝑓 of π‘₯, the solutions to the equation 𝑓 of π‘₯ equals zero correspond to the values of the π‘₯-intercepts if they exist. If there are no π‘₯-intercepts, it follows that there are no solutions to the equation 𝑓 of π‘₯ equals zero. Now in this case, we’ve been given the graph of the function 𝑓 of π‘₯ equals π‘₯ squared minus two π‘₯ plus three and asked to solve 𝑓 of π‘₯ equals zero or, in other words, π‘₯ squared minus two π‘₯ plus three equals zero.

But if we look carefully, we see that there are no locations where the graph, which is the green plot here, intersects the π‘₯-axis. And so, there are no real solutions to the equation 𝑓 of π‘₯ equals zero. In order to use set notation, we need to find a way to show that there are no values inside the set. So we use the notation shown. It’s the null set or empty set.

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