The quadratic formula
The discriminant is the part of the quadratic formula underneath the square root symbol: b²-4ac. The discriminant tells us whether there are two solutions, one solution, or no solutions.
The quadratic formula
The discriminant is the term underneath the square root in the quadratic formula and tells us the number of solutions to a quadratic equation. If the discriminant is positive, we know that we have 2 solutions. If it is negative, there are no solutions and if the discriminant is equal to zero, we have one solution. The discriminant is calculuated by squaring the "b" term and subtracting 4 times the "a" term times the "c" term.
The discriminant is a really handy tool when you think you're getting a weird answer. Here's why. The discriminant tells you how many solutions there are to quadratic equation or how many x intercepts there are for a parabola. It doesn't tell you what those numbers are like what the x intercept values are, it just tells you how many of them there should be.
And it sounds like that's not useful but it actually is especially when you're checking your work.
So here is what it looks like. The discriminant is the formula b squared minus 4ac remembering that a, b and c are the coefficients of your quadratic in standard form. It tells you the number of solutions to a quadratic equation. If the discriminant is greater than zero, there are two solutions. If the discriminant is less than zero, there are no solutions and if the discriminant is equal to
zero, there is one solution.
This is something you just kind of have to memorize. It goes hand in hand with the quadratic formula. So if you guys have learned that, this will make a lot of sense. If you haven't learnt the quadratic formula yet you'll probably learn it tomorrow in math class. Just know that, what you're looking at is whether or not b squared minus 4ac is greater than zero, less than zero or equal to zero. And it tells me how many answers I should have. It doesn't tell me what
the answers are just how many of them I should have in order to get the problem correct.
In algebra, the discriminant of a polynomial is a polynomial function of its coefficients, which allows deducing some properties of the roots without computing them.1
You are probably aware of the well-known formula of the discriminant for the quadratic polynomial , which is , and use this formula to compute the roots.
However, the discriminant actually allows us to deduce some properties of the roots without computing them. In the case of a quadratic polynomial, it is zero if – and only if – the polynomial has a double root. It is positive if the polynomial has two real roots, and it is negative if roots are complex.
The calculator below computes the discriminant, and you can find a bit more theory on discriminants immediately underneath it.
The discriminant of the quadratic polynomial
Discriminant
The discriminant for a polynomial of degree n: can be defined either in terms of the quotient of the resultant or in terms of the roots.
In terms of the roots, the discriminant is equal to
Technically, one can derive the formula for the quadratic equation without knowing anything about the discriminant. Then, if you plug derived expressions for the roots into the definition above, you will end up with the .
In terms of the quotient of the resultant, the discriminant is equal to
where Res is the resultant of A and the first derivation of A'. The resultant, in short, is the
determinant of the Sylvester matrix of A and A'.
In the case of a quadratic polynomial, the A is and the A' is . If you write down the Sylvester matrix for these two polynomials and derive the determinant, you will again end up with the .
Higher degree discriminant computation
Using the second definition, you can derive formulas for a polynomial of higher degrees (the link below has formulas for degree 3 and degree 4), but they are quite complex.
OEIS sequence A007878 lists 5 terms for polynomials of a degree
of 3; 16 terms for a degree of 4; 59 terms for a degree of 5; and finally 3,815,311 terms for polynomials of a degree of 12.
The calculator below computes the discriminant of a higher degree polynomial from the resultant of a polynomial and its derivative
Discriminant
The polynomial coefficients, space separated, in order from higher term degree to lower