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Aptitude Topics

Quadratic Equations

A quadratic equation is of the form ax² + bx + c = 0 where a ≠ 0. Every quadratic equation has exactly two roots (real or complex), and those roots reveal key properties of the equation.

Fundamental Principles

Quadratic Equation

An equation of degree 2 written as ax² + bx + c = 0, where a, b, c are constants and a ≠ 0.

Discriminant

The value D = b² - 4ac. If D > 0: two distinct real roots. If D = 0: two equal real roots. If D < 0: no real roots (complex roots).

Vieta's Formulas

For ax² + bx + c = 0 with roots α and β: Sum of roots α + β = -b/a. Product of roots αβ = c/a.

Essential Formulation Tips

  • Try factoring first — it's the fastest method when it works.
  • Use the quadratic formula when factoring is not obvious.
  • Always check the discriminant to know what kind of roots to expect.
  • Vieta's formulas let you find sum and product of roots without solving.

Shortcut Execution Techniques

  • Sum of roots = -b/a; Product of roots = c/a.
  • If roots are equal and opposite → b = 0.
  • If product of roots is negative → roots have opposite signs.
  • If sum of roots is 0 → roots are equal and opposite.

Contextual Inquiries (FAQs)

Q: What is the quadratic formula?

A: x = (-b ± √(b² - 4ac)) / 2a.

Q: What does it mean if D = 0?

A: The quadratic has two equal (repeated) roots: x = -b/2a.

Q: Can a quadratic equation have more than 2 roots?

A: No. A quadratic equation always has exactly 2 roots (counting multiplicity).