What It Means for a Quadratic to Have One Solution
A quadratic equation has exactly one solution when its discriminant is zero. In standard form ax^2 + bx + c = 0, the discriminant Δ = b^2 − 4ac determines the number and type of solutions. When Δ = 0, the quadratic touches the x-axis at a single point, producing one repeated real root. This occurs because the parabola’s vertex lies exactly on the x-axis. The one solution is often called a double root or repeated root, written as x = −b / (2a). Understanding this condition clarifies when two solutions merge into one and how it relates to the graph’s geometry.
Why the Discriminant Controls the Number of Solutions
The Role of b^2 − 4ac
The discriminant Δ = b^2 − 4ac is the key quantity under the square root in the quadratic formula:
- If Δ > 0, there are two distinct real solutions.
- If Δ = 0, there is exactly one real solution (a repeated root).
- If Δ
When Δ = 0, the square root term vanishes, so the quadratic formula simplifies to x = −b / (2a). This single value is the x-coordinate of the vertex and the sole x-intercept of the parabola. The condition b^2 = 4ax is both necessary and sufficient for one real solution in the real number system.
Worked Example: Quadratic with One Solution
Consider the quadratic equation x^2 − 6x + 9 = 0. Identify coefficients a = 1, b = −6, and c = 9. Compute the discriminant:
Δ = b^2 − 4ac = (−6)^2 − 4(1)(9) = 36 − 36 = 0.
Because Δ = 0, there is exactly one real solution. Apply the quadratic formula:
x = (−b ± √Δ) / (2a) = (6 ± 0) / 2 = 3.
The equation has a repeated root at x = 3. In factored form, this is (x − 3)^2 = 0, confirming the double root. The parabola y = x^2 − 6x + 9 touches the x-axis at (3, 0) and does not cross it.
Relating the Example to the Graph
Vertex as the Single x-Intercept
For y = x^2 − 6x + 9, the vertex lies at x = −b / (2a) = 3. Substituting gives y = 0, so the vertex is (3, 0). Because the parabola opens upward (a > 0) and the vertex sits on the x-axis, the graph touches the axis at exactly one point. This geometric view aligns with the algebraic conclusion of one solution. If the vertex were above the axis (a > 0 and y_vertex > 0) or below (a
Common Misconceptions and Clarifications
It is sometimes mistakenly believed that a quadratic always has two solutions. In the real number system, the count of x-intercepts can be 0, 1, or 2, corresponding to Δ 0. Another misconception is that a double root is not a "real" solution; in fact, it is a real solution with multiplicity two, important in algebra and calculus. Also, while completing the square or factoring can reveal the repeated root, the discriminant offers a quick diagnostic without solving fully.
Summary of Key Conditions
| Condition | Value | Number of Real Solutions | Graph Behavior |
|---|---|---|---|
| Discriminant Δ > 0 | Positive | Two distinct real solutions | Parabola crosses x-axis twice |
| Discriminant Δ = 0 | Zero | One real solution (repeated root) | Parabola touches x-axis at vertex |
| Discriminant Δ | Negative | No real solutions (two complex) | Parabola does not intersect x-axis |
How to Identify and Construct Examples
To create a quadratic with one solution, choose any real number r for the repeated root and any nonzero a. The equation a(x − r)^2 = 0 expands to ax^2 − 2arx + ar^2 = 0. In this form, b = −2ar and c = ar^2, which guarantees b^2 = 4ac and Δ = 0. For example, choosing a = 2 and r = −1 yields 2(x + 1)^2 = 0, or 2x^2 + 4x + 2 = 0. Verifying: Δ = 4^2 − 4(2)(2) = 16 − 16 = 0, confirming one solution at x = −1. This constructive approach helps reinforce the discriminant condition.