If B2 4ac 0 Then Roots Are

If B² - 4AC = 0, Then Roots Are: A Deep Dive into Quadratic Equations

Quadratic equations are fundamental in algebra, appearing in fields ranging from physics to economics. At their core, these equations take the form ax² + bx + c = 0, where a, b, and c are constants, and a ≠ 0. The solutions to these equations, known as roots, reveal critical information about the behavior of parabolic graphs. One of the most pivotal tools for analyzing quadratic equations is the discriminant, a value calculated as B² - 4AC. This article explores what happens when the discriminant equals zero, unraveling the mathematical and practical implications of this condition.

Understanding the Discriminant: The Key to Roots

The discriminant, B² - 4AC, is derived from the quadratic formula:
x = [-B ± √(B² - 4AC)] / (2A).

This formula provides the roots of the equation, but the discriminant’s value dictates the nature of these roots:

• If B² - 4AC > 0: Two distinct real roots.
• If B² - 4AC = 0: One real root (a repeated root).
• If B² - 4AC < 0: Two complex conjugate roots.

Today, we focus on the second case: when the discriminant equals zero. This scenario leads to a unique situation where the quadratic equation has exactly one real solution, often referred to as a repeated root or double root.

What Happens When B² - 4AC = 0?

When the discriminant is zero, the quadratic formula simplifies dramatically. The ±√(B² - 4AC) term becomes zero, leaving:
x = -B / (2A).

This means the equation has a single solution, repeated twice. For example, consider the equation x² - 4x + 4 = 0. Here, A = 1, B = -4, and C = 4. Calculating the discriminant:
(-4)² - 4(1)(4) = 16 - 16 = 0.

Substituting into the quadratic formula:
x = -(-4) / (2·1) = 4 / 2 = 2.

Thus, the root x = 2 is repeated, meaning the parabola touches the x-axis at this point but does not cross it.

Geometric Interpretation: A Parabola Tangent to the X-Axis

The condition B² - 4AC = 0 has a striking geometric interpretation. The graph of a quadratic equation is a parabola. When the discriminant is zero:

1. The parabola touches the x-axis at exactly one point (the vertex).
2. The vertex lies on the x-axis, making it the only intersection point between the parabola and the axis.

For instance, the equation y = x² - 6x + 9 simplifies to y = (x - 3)². Its vertex is at (3, 0), and the parabola opens upward, just grazing the x-axis at x = 3. This visualizes why the root is repeated—it’s the sole point of contact.

Real-World Applications: Why Repeated Roots Matter

Repeated roots aren