Understanding Quadratic Equations with a Discriminant Equal to Zero

The discriminant, b2 - 4ac, is a key component in the analysis of quadratic equations, ax2 bx c 0. It provides crucial information about the nature of the roots of the equation. When the discriminant is equal to zero, it signifies that the quadratic equation has two identical real roots. This article explains why this is possible and its implications in solving such equations and in the broader applications of the discriminant.

What Happens When the Discriminant is Zero?

For a quadratic equation of the form ax2 bx c 0, the discriminant is calculated as d b2 - 4ac. The value of the discriminant can give us valuable insights into the roots of the equation. If the discriminant is equal to zero, i.e., d 0, then the quadratic equation has two identical real roots, which means there are two distinct but exactly equal solutions. This situation occurs because the quadratic formula, x frac{-b pm sqrt{b^2 - 4ac}}{2a}, results in the same value for both roots.

Example and Solution

Let's consider a simple example where the discriminant is zero:

Example: Solve the quadratic equation x2 - 2x 1 0.

a 1, b -2, c 1 Discriminant d b2 - 4ac (-2)2 - 4(1)(1) 4 - 4 0

Using the quadratic formula:

i. x frac{-b pm sqrt{b2 - 4ac}}{2a} frac{-(-2) pm sqrt{(-2)2 - 4(1)(1)}}{2(1)} frac{2 pm sqrt{0}}{2} frac{2 pm 0}{2} 1

Thus, the solution is x 1, which is a repeated root.

Mathematical Justification

The condition d 0 implies that the roots of the quadratic equation are equal. This can be shown as:

From the quadratic formula, x frac{-b pm sqrt{b2 - 4ac}}{2a}, if d b2 - 4ac 0, then sqrt{b2 - 4ac} 0. This results in x frac{-b pm 0}{2a} frac{-b}{2a}, which is a single root.

Thus, the case where d 0 leads to a repeated root, meaning the quadratic equation has two identical real roots.

Application in Perfect Squares

Another way to illustrate this is by considering a quadratic equation that can be expressed as a perfect square. For example:

Example: Consider the equation x2 - 4x 4 0.

a 1, b -4, c 4

The discriminant is:

i. b2 - 4ac (-4)2 - 4(1)(4) 16 - 16 0

Factoring the equation, we get:

(x - 2)2 0

Solving for x gives:

x 2, which is a repeated root.

This example demonstrates that a perfect square can result in a discriminant of zero, leading to a repeated root.

Implications Beyond Quadratic Equations

The concept of the discriminant being zero also extends beyond the realm of quadratic equations. In the context of Partial Differential Equations (PDE), the discriminant plays a crucial role in classifying the type of equation:

P b2 - 4ac If P , the equation is elliptic. If P 0, the equation is parabolic. If P > 0, the equation is hyperbolic.

In the case of a parabolic equation, the discriminant equals zero, indicating that the equation is a Parabolic PDE. An example of such an equation is:

y uxx - 4 uxy - 4x uyy 0

Here, the discriminant P 16 - 16xy, and when xy 1, the discriminant is zero, indicating a parabolic type of PDE.

By understanding the implications of the discriminant being zero, we can gain deeper insights into both the solutions of quadratic equations and the classification of PDEs.