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. In practice, one of the most central 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).

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This means the equation has a single solution, repeated twice. Here's the thing — for example, consider the equation x² - 4x + 4 = 0. Day to day, 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. In real terms, the parabola touches the x-axis at exactly one point (the vertex). And 2. The vertex lies on the x-axis, making it the only intersection point between the parabola and the axis.

Take this: 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