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Question Video: Using the Sign of the Discriminant to Determine the Number of Times a Quadratic Crosses the ๐’™-Axis Mathematics • 9th Grade

Find the discriminant of the quadratic equation 2๐‘ฅยฒ + 3๐‘ฅ + 4 = 0. How many real roots does the equation 2๐‘ฅยฒ + 3๐‘ฅ + 4 = 0 have? Hence, decide how many times the graph of ๐‘ฆ = 2๐‘ฅยฒ + 3๐‘ฅ + 4 will cross the ๐‘ฅ-axis.

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Video Transcript

Find the discriminant of the quadratic equation two ๐‘ฅ squared plus three ๐‘ฅ plus four equals zero. How many real roots does the equation two ๐‘ฅ squared plus three ๐‘ฅ plus four equals zero have? Hence, decide how many times the graph of ๐‘ฆ equals two ๐‘ฅ squared plus three ๐‘ฅ plus four will cross the ๐‘ฅ-axis.

Letโ€™s begin by reminding ourselves what it means to find the discriminant of a quadratic equation. Suppose weโ€™re given a quadratic equation of the form ๐‘Ž๐‘ฅ squared plus ๐‘๐‘ฅ plus ๐‘ equals zero, where ๐‘Ž is not equal to zero. The discriminant that we define using the ฮ” symbol is the part of the quadratic formula that sits inside the square root. Itโ€™s ๐‘ squared minus four ๐‘Ž๐‘. And the discriminant is really useful as weโ€™ll see in a moment, as it tells us the number of real roots of our equation.

Before we move on to that though, letโ€™s just define the discriminant of the equation two ๐‘ฅ squared plus three ๐‘ฅ plus four equals zero. ๐‘Ž is the coefficient of ๐‘ฅ squared, so itโ€™s two. The value of ๐‘ is the coefficient of ๐‘ฅ, so itโ€™s three. And ๐‘ is the constant term; itโ€™s four. This means the discriminant is three squared minus four times two times four. Three squared is nine, and four times two times four is 32. So our discriminant is nine minus 32, which is negative 23. So the discriminant of our equation is negative 23.

The next part of this question asks us to find the number of real roots that our quadratic has. When we talk about roots, weโ€™re talking about the number of solutions to our equation. And the discriminant can tell us how many roots we have. If we think about substituting our values into the quadratic formula, if the discriminant, the part inside the square root, is positive, we know that we get the square root of a positive number, which is a real number. And so when the discriminant is positive, when itโ€™s greater than zero, we get two real roots. We get two solutions to our equation.

If, however, the discriminant is equal to zero, weโ€™re taking the square root of zero, which is zero. This means we get negative ๐‘ divided by two ๐‘Ž as the only real solution to our equation. And so if the discriminant is zero, we get one real root. If, however, the discriminant is negative, weโ€™re taking the square root of a negative number, which is not a real number. And so if the discriminant is negative, we get no real roots. Now, of course, our discriminant is negative 23, which is less than zero. Since our discriminant is negative, there must be no real roots to the equation two ๐‘ฅ squared plus three ๐‘ฅ plus four equals zero. And so the answer to the second part of this question is zero.

Finally, letโ€™s use this information to determine the number of times the graph of ๐‘ฆ equals two ๐‘ฅ squared plus three ๐‘ฅ plus four crosses the ๐‘ฅ-axis. And we can use the graph of the general quadratic function, where the coefficient of ๐‘ฅ squared is positive, to establish whatโ€™s happening here. The ๐‘ฅ-axis can equivalently be expressed as the line ๐‘ฆ equals zero. This means if we set the quadratic function equal to zero and then solve for ๐‘ฅ, we find the locations of the ๐‘ฅ-intercepts. We find out where the graph of the function crosses the line ๐‘ฆ equals zero.

This means then, in turn, that the number of real roots we have tells us equivalently the number of times our graph passes through the ๐‘ฅ-axis. Since we determined that the equation two ๐‘ฅ squared plus three ๐‘ฅ plus four equals zero has no real roots, we can equivalently say that the graph of ๐‘ฆ equals two ๐‘ฅ squared plus three ๐‘ฅ plus four does not cross the ๐‘ฅ-axis at all. It crosses it zero times. Since the coefficient of ๐‘ฅ squared is positive, it might look a little something like this. Every output to this function is positive.

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