What does it mean if a discriminant is negative?

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The quadratic formula and the discriminant
What does a negative discriminant tell you about the graph?
When the discriminant is negative what are the roots?
What does finding the discriminant tell us?
Is the discriminant is negative then the quadratic has?

José F.

Mar 12, 2018

If you have a polynomial of the type #ax^2+bx+c#, the discriminant is #b^2-4ac#

Explanation:

Having a negative discriminant means that #b^2-4ac<0#, and the polynominal doesn't have real solutions.

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What this means is that $m^2-2m+9$ has no real roots, i.e., it is always either positive or negative (because polynomials are continuous, there are no "gaps", so for the function to switch sign it would need to cross the x-axis and would have a root). To find out which, we evaluate at $m=1$:

$$1^2-1 \cdot 2 + 9 =1-2+9=8$$

This is positive, and therefore we can say that $m^2-2m+9$ is always positive.

This means $b^2-4ac$ of the original function is always positive. This tells you that your original function, no matter what the value of $m$ is, has two real roots.

In general, if the discriminant of a function $f(x)$ is a function $g(m)$, if the discriminant of $g(m)$ is always negative, then $f(x)$'s discriminant ($g(m)$) is either always negative for any value of $m$ or always positive. You can determine which by evaluating $g(0)$ or any similarly easy evaluation. If the discriminant of $f(x)$ is always negative for any value of $m$, it means that $f$ is guaranteed to have no real roots. If the discriminant of $f(x)$ is always positive, it means that $f$ is guaranteed to have two real roots.

On the opposite side, if $g(m)$'s discriminant is positive, then the number of real roots of $f$ varies depending on the value of $m$ chosen.

The quadratic formula and the discriminant

The quadratic formula was covered in the module Algebra review. This formula gives solutions to the general quadratic equation $ax^2+bx+c=0$, when they exist, in terms of the coefficients $a,b,c$. The solutions are

\[ x = \dfrac{-b+\sqrt{b^2-4ac}}{2a}, \qquad x = \dfrac{-b-\sqrt{b^2-4ac}}{2a}, \]

provided that $b^2-4ac \geq 0$.

The quantity $b^2-4ac$ is called the discriminant of the quadratic, often denoted by $\Delta$, and should be found first whenever the formula is being applied. It discriminates between the types of solutions of the equation:

$\Delta > 0 $ tells us the equation has two distinct real roots
$\Delta = 0 $ tells us the equation has one (repeated) real root
$\Delta < 0 $ tells us the equation has no real roots.

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, Interactive 2

Exercise 4

For what values of $k$ does the equation $(4k+1)x^2-6kx+4=0$ have one real solution?

A quadratic expression which always takes positive values is called positive definite, while one which always takes negative values is called negative definite.

Quadratics of either type never take the value 0, and so their discriminant is negative. Furthermore, such a quadratic is positive definite if $a > 0$, and negative definite if $a < 0$.

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Example

Show that the quadratic expression $4x^2-8x+7$ always takes positive values for any value of $x$.

Solution

In this case, $a=4$, $b=-8$ and $c=7$. So

\[ \Delta = (-8)^2 - 4\times 4 \times 7 = -48 < 0 \]

and $a=4 > 0$. Hence the quadratic is positive definite.

Exercise 5

For what values of $k$ does the equation $(4k+1)x^2-2(k+1)x+(1-2k)=0$ have one real solution? For what values of $k$, if any, is the quadratic negative definite?

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What does a negative discriminant tell you about the graph?

A negative discriminant means that the graph of the quadratic equation does not touch the 𝑥-axis. If the quadratic has a positive coefficient of 𝑥² (a > 0), the graph is concave up and the entirety of the graph is above the 𝑥-axis. All outputs of the graph are positive.

When the discriminant is negative what are the roots?

If the discriminant of a quadratic equation is negative, then the roots of the equation are not real.

What does finding the discriminant tell us?

The discriminant gives us important information about the quadratic equation. First, it can tell us how many solutions the quadratic equation has. It can also tell us how many times the graph crosses the x-axis and if the solutions are real or complex. But the discriminant is just a number!