How Many Solutions When Discriminant Is 0?

When the discriminant of a quadratic equation is 0, there is exactly one real solution. This solution is often referred to as a "repeated" or "double" root because it arises from two identical factors.

Understanding the Discriminant

The discriminant is a crucial part of the quadratic formula, which is used to solve equations of the form ax² + bx + c = 0. It is the expression b² - 4ac. The value of the discriminant determines the nature and the number of solutions a quadratic equation will have.

• b² - 4ac > 0 (Positive): There are two distinct real solutions.
• b² - 4ac = 0 (Zero): There is one real, repeated solution.
• b² - 4ac < 0 (Negative): There are two complex conjugate solutions (no real solutions).

Why Zero Discriminant Means One Solution

When the discriminant is 0, the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a simplifies significantly. Since sqrt(0) is 0, the ± part of the formula becomes ± 0. This means that [-b + 0] / 2a and [-b - 0] / 2a both result in the same value: x = -b / 2a.

This single result indicates that the quadratic equation has only one unique real solution, even though it technically comes from two identical mathematical operations.

Practical Example

Consider the quadratic equation: x² - 6x + 9 = 0

Here, a = 1, b = -6, and c = 9.

Let's calculate the discriminant:
Discriminant = b² - 4ac
Discriminant = (-6)² - 4(1)(9)
Discriminant = 36 - 36
Discriminant = 0

Since the discriminant is 0, we expect one real, repeated solution.
Using the quadratic formula:
x = [-(-6) ± sqrt(0)] / 2(1)
x = [6 ± 0] / 2
x = 6 / 2
x = 3

The equation x² - 6x + 9 = 0 can also be factored as (x - 3)(x - 3) = 0, which clearly shows x = 3 as the single, repeated solution.

Graphical Interpretation

Graphically, a quadratic equation represents a parabola.

• If the discriminant is positive, the parabola intersects the x-axis at two distinct points, representing two real solutions.
• If the discriminant is negative, the parabola does not intersect the x-axis at all, indicating no real solutions (only complex ones).
• If the discriminant is zero, the parabola touches the x-axis at exactly one point. This point is the vertex of the parabola, and its x-coordinate is the single, repeated real solution.

Summary of Solution Types

The following table summarizes how the value of the discriminant influences the nature of the solutions for a quadratic equation:

Understanding the discriminant is fundamental for analyzing quadratic equations and predicting the nature of their solutions without fully solving them. For more details on quadratic equations and the discriminant, you can explore resources like Khan Academy's section on the discriminant (external link).