zaro

What is the Nature of the Roots of a Quadratic Equation Determined by the Value of its Discriminant?

Published in Quadratic Roots Nature 3 mins read

The nature of the roots of a quadratic equation is fundamentally determined by the value of its discriminant, which is the expression b² - 4ac. For a standard quadratic equation in the form ax² + bx + c = 0 (where a ≠ 0), the discriminant provides critical insight into whether the roots are real or imaginary, and if they are equal or unequal.

Understanding the Discriminant

The discriminant, often denoted by the symbol delta (Δ), plays a pivotal role because it is the term under the square root in the quadratic formula (x = [-b ± sqrt(b² - 4ac)] / 2a). Its value dictates the characteristics of the solutions to the quadratic equation.

Cases for the Nature of Roots

The nature of the roots can be categorized into several distinct cases based on the value of the discriminant:

Value of Discriminant (b² - 4ac) Nature of Roots
b² - 4ac > 0 Real and unequal
b² - 4ac = 0 Real and equal
b² - 4ac < 0 Unequal and Imaginary
b² - 4ac > 0 (is a perfect square) Real, rational, and unequal
b² - 4ac > 0 (is not a perfect square) Real, irrational, and unequal

Let's delve deeper into each scenario:

1. Discriminant is Greater Than Zero (b² - 4ac > 0)

When the discriminant is a positive value, the quadratic equation has two distinct real roots. This means the parabola corresponding to the quadratic equation intersects the x-axis at two different points.

  • Real and Unequal: The roots are distinct real numbers.
  • Perfect Square Condition: If the positive discriminant is also a perfect square (e.g., 4, 9, 25), then the square root of the discriminant will be an integer. This results in roots that are real, rational, and unequal. Rational numbers can be expressed as a fraction of two integers.
  • Not a Perfect Square Condition: If the positive discriminant is not a perfect square (e.g., 2, 7, 10), then the square root of the discriminant will be an irrational number. This leads to roots that are real, irrational, and unequal. Irrational numbers cannot be expressed as a simple fraction.

2. Discriminant is Equal to Zero (b² - 4ac = 0)

If the discriminant is exactly zero, the quadratic equation has exactly one real root, which is often referred to as a repeated root or two equal real roots. In graphical terms, the parabola touches the x-axis at exactly one point (its vertex lies on the x-axis).

  • Real and Equal: The two roots are identical real numbers. This occurs because the ± sqrt(b² - 4ac) term in the quadratic formula becomes ± 0, effectively leaving only one solution, -b / 2a.

3. Discriminant is Less Than Zero (b² - 4ac < 0)

When the discriminant is a negative value, the quadratic equation has no real roots. Instead, it has two complex (or imaginary) roots. Graphically, the parabola does not intersect the x-axis at all; it either lies entirely above or entirely below it.

  • Unequal and Imaginary: The roots are complex conjugates of each other. This means they are of the form p + qi and p - qi, where i is the imaginary unit (sqrt(-1)). Since the square root of a negative number is imaginary, the roots will contain an imaginary component.

Understanding the discriminant is crucial for predicting the nature of solutions to quadratic equations without actually solving them.