Definite and indefinite quadratics
We have seen how the value of the discriminant can tell us about the number of real solutions to a quadratic equation. Recall that, for a quadratic equation in the general form $ax^2+bx+c=0$ax2+bx+c=0, the discriminant is given by $\Delta=b^2-4ac$Δ=b2−4ac. Here is a graphical summary of the range of values that the discriminant can take:
Let's turn our focus to that last result, when the value of the discriminant is less than zero. There is no real number that returns a negative value when squared, so there are no real roots to these equations, which means their function graph has no $x$x-intercepts.
When the discriminant is less than zero, the graphs of these functions will lie either entirely above or entirely below the $x$x-axis.
Look at the upper parabola in the image above. For any value of $x$x, the corresponding value of $y$y will be positive. We call this type of quadratic positive definite. In this case, note also that the value of $a$a in the equation $y=ax^2+bx+c$y=ax2+bx+c will be greater than zero (this is a concave up parabola).
Similarly, the parabola on the bottom has only negative $y$y values, and so this type of quadratic is negative definite. The value of the coefficient $a$a in this case will be less than zero (this is a concave down parabola).
A quadratic equation that does cross the $x$x-axis has some parts that lie above, and some parts that lie below the axis. They are not definitely all positive or all negative, and so we refer to these types of quadratics as indefinite. A quadratic equation with discriminant greater than zero will be indefinite, regardless of the value of the coefficient $a$a.
Here is a summary of these ideas.
| Value of $\Delta$Δ | Value of $a$a | |
|---|---|---|
| Positive definite | $\Delta<0$Δ<0 | $a>0$a>0 |
| Negative definite | $\Delta<0$Δ<0 | $a<0$a<0 |
| Indefinite | $\Delta>0$Δ>0 | $a<0$a<0 or $a>0$a>0 |
Examples
Question 1
Consider the quadratic equation $y=7x^2+8x+3$y=7x2+8x+3.
Evaluate its discriminant.
By referring to the discriminant and to the coefficient of $x^2$x2, one can conclude that the function is:
Negative definite
AIndefinite
BPositive definite
CWhich graph represents the quadratic equation?
Loading Graph...ALoading Graph...BLoading Graph...CLoading Graph...D
Question 2
Consider the quadratic equation $y=3x^2+18x+m$y=3x2+18x+m.
Find the discriminant in terms of $m$m.
For what values of $m$m is the function positive definite?
For what values of $m$m is the function indefinite?