The nature of roots of quadratic equations

The Quadratic Formula and Nature of Roots

In approximately 1025 A.D., the renowned Indian mathematician Sridharacharya presented a systematic approach—now known as the quadratic formula—for solving equations of the form: \( ax^2 + bx + c = 0 \), where a ≠ 0.

The Quadratic Formula:
The solutions to the quadratic equation \( ax^2 + bx + c = 0 \) can be found using the formula:
\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) Here, the term under the square root, \( D = b^2 - 4ac \), is called the discriminant.

Nature of the Roots Based on the Discriminant:

• If \( D > 0 \): The equation has two distinct real roots.
• If \( D = 0 \): The equation has two equal real roots (repeated root).
• If \( D < 0 \): The equation has two complex roots (no real roots).

Example: Examine Which of the Following Quadratic Equations Have Real Roots

I) Equation: \( x^2 - 5x + 6 = 0 \)

By comparing with the standard quadratic form \( ax^2 + bx + c = 0 \):
We identify \( a = 1 \), \( b = -5 \), and \( c = 6 \).

Discriminant:
\( D = (-5)^2 - 4(1)(6) = 25 - 24 = 1 \)

Since \( D > 0 \), the equation has two distinct real roots.

II) Consider the quadratic equation: \( 2x^2 - 4x + 5 = 0 \)

Here, \( a = 2 \), \( b = -4 \), \( c = 5 \)

Discriminant:
\( D = (-4)^2 - 4(2)(5) = 16 - 40 = -24 \)

Since \( D < 0 \), the equation has no real roots (it has complex roots).

Conclusion:

• Equation I has real roots.
• Equation II has no real roots (complex roots).