Conditions and Properties of Quadratic Equations: Real, Complex, and Infinite Roots

The quadratic equation is a fundamental aspect of algebra and plays a crucial role in various fields such as engineering, physics, and mathematics. The standard form of a quadratic equation is given by:

Standard Form of a Quadratic Equation

y ax^2 bx c

where a, b, and c are constants, and a ≠ 0. The roots of the quadratic equation ax^2 bx c 0 can be found using the quadratic formula:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Deriving the Roots from the Standard Form

The quadratic equation can be rewritten in the form:

x^2 frac{b}{a}x frac{c}{a} 0

Completing the square, we can express it as:

x^2 frac{b}{a}x left(frac{b}{2a}right)^2 - left(frac{b}{2a}right)^2 frac{c}{a} 0

This can be simplified to:

left(x frac{b}{2a}right)^2 frac{b^2 - 4ac}{4a^2}

Taking the square root of both sides, we get:

x frac{b}{2a} frac{pm sqrt{b^2 - 4ac}}{2a}

Finally, solving for x, we obtain:

x frac{-b pm sqrt{b^2 - 4ac}}{2a}

Condition for Real and Distinct Roots

For the roots to be real and distinct, the discriminant D b^2 - 4ac must be greater than zero (D > 0). This can be written as:

D>0 → Roots are real and distinct.

Condition for Real and Equal Roots

For the roots to be real and equal, the discriminant must be zero (D 0). This condition can be written as:

D0 → Roots are real and equal.

Condition for Complex Roots

For the roots to be complex conjugates, the discriminant must be less than zero (D ). This condition can be written as:

D → Roots are complex conjugates.