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Welcome to our Math lesson on The Meaning of Discriminant, this is the second lesson of our suite of math lessons covering the topic of The Quadratic Formula, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.
Not all quadratic equations have two distinct roots. Some quadratic equations may have a single root. For example, the quadratic equation
has a single root, because the left part represents the expanded form of the first special algebraic identity
Therefore, we can write our equation as
It is obvious that this equation has a single root, x = -2, as only for this value of the variable x the equation is true.
Moreover, there are some quadratic equations that have no solution in the set of real numbers (we will see towards the end of this course that these quadratic equations are solved in the set of complex numbers).
We can identify which of the above types a quadratic equation belongs to by looking at the part of the quadratic formula inside the square root. This is because when the expression inside the root is positive, we obtain two different values after calculating the root and therefore, two different solutions for the equation. On the other hand if the expression inside the square root is zero we obtain two equal solutions, as the root is zero. Last, if the root is negative we cannot continue with the solution as we cannot calculate the square root of a negative number.
From all discussed above, it is clear that the expression inside the root makes the distinction between various cases of quadratic equations in regard to the number of roots is has. We call this part of the solution as discriminant, Δ. Thus, we have
Thus, summarizing the above findings, we can say that:
Find the number of roots in the quadratic equations below without making the calculations.
We have just to check the sign of the discriminant in order to know the number of roots in a given quadratic equation.
We can obtain useful information about the original equation quadratic equation by studying the discriminant. Let's consider an example to clarify this point.
The quadratic equation x2 - mx + 9 = 0 has two equal roots. What is/are the value(s) of m?
From theory it is known that when a quadratic equation has two equal roots (otherwise we say it has a single root), then its discriminant is zero. We have a = 1, b = -m and c = 9. Hence, from the formula of discriminant Δ, we have
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