The Discriminant
Use the discriminant D = b² − 4ac to determine the number and nature of solutions before solving. Understand and interpret D > 0 (two real roots), D = 0 (one real root), and D < 0 (no real roots) in graphical terms.
A lesson of Algebra 1, in Quadratic Functions, about 30 minutes.
By the end of this lesson, you will be able to compute $b^2 - 4ac$, read off how many solutions an equation has before solving it, say what that looks like on a graph, and tell whether a quadratic will factor.
One Number, Computed First
$$D = b^2 - 4ac$$
The part under the root in the quadratic formula, on its own. Computing it before anything else tells you what kind of answer to expect, which means you know whether the work ahead is worth starting.
$$D > 0 \quad \text{two real solutions}$$
$$D = 0 \quad \text{one real solution, repeated}$$
$$D < 0 \quad \text{no real solutions}$$
Why the sign is the whole story. The formula puts $\sqrt{D}$ into a $\pm$. A positive $D$ gives a real root, and adding it or subtracting it produces two different answers. A zero $D$ makes the $\pm$ do nothing, so both branches give $-\frac{b}{2a}$. A negative $D$ has no real square root at all, so neither branch exists.
What It Looks Like on a Graph
The solutions of $ax^2 + bx + c = 0$ are where the parabola meets the $x$-axis, so $D$ is really a statement about position.
| $D$ | Solutions | The parabola |
|---|---|---|
| $D > 0$ | two | crosses the axis twice |
| $D = 0$ | one | touches the axis at its vertex |
| $D < 0$ | none | misses the axis entirely |
$D = 0$ is the vertex sitting exactly on the axis, which is why the solution is "repeated": the two crossing points have slid together into one.
$D < 0$ means the whole parabola is on one side of the axis: above it if it opens up, below if it opens down. There is nothing to find because it never gets there.
That picture makes the arithmetic memorable. Three cases, three positions, and you can sketch any of them from the sign of one number.
Sort It: How Many Real Solutions
Compute $D$ and read the sign.
Example 1: Computing It Carefully
Here is every step on one equation, worked all the way through. $$2x^2 - 5x + 3 = 0$$
- What is a? 2
- What is b squared? 25
- What is 4ac? 24
- What is the discriminant? 1
- D is positive, so how many real solutions are there? 2
A Perfect Square Means It Factors
$D$ answers a second question that the three cases do not.
$$D \text{ is a perfect square} \iff \text{the quadratic factors over the integers}$$
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