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∫₀¹ x² dx = ⅓
e + 1 = 0
a² + b² = c²
∇ × F = curl F
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The Quadratic Formula

For any quadratic equation of the form ax² + bx + c = 0, the solutions are:

x = (-b ± √(b² - 4ac)) / (2a)

The expression under the square root, b² - 4ac, is called the discriminant. It tells us:

  • b² - 4ac > 0: Two distinct real solutions
  • b² - 4ac = 0: Exactly one real solution (repeated root)
  • b² - 4ac < 0: Two complex conjugate solutions
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The Fundamental Theorem of Calculus

This theorem connects differentiation and integration — the two main operations of calculus:

∫ₐᵇ f(x) dx = F(b) - F(a)

Where F is any antiderivative of f, meaning F'(x) = f(x). This tells us that:

  • Integration and differentiation are inverse processes
  • Definite integrals can be evaluated using antiderivatives
  • The net area under a curve equals the difference of antiderivative values
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