Quadratic Equations and Theory of Equations

Theory of Equations and Advanced Topics

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Theory of Equations and Advanced Topics

Theory of Equations and Advanced Topics

This lesson covers the following key concepts:

  • Equations reducible to quadratic form (biquadratic)
  • Symmetric functions of roots
  • Transformation of equations
  • Common roots of two quadratic equations
  • Quadratic inequalities and wavy curve method
  • Maximum and minimum problems
  • Applications to JEE problems

Important Formulas

  • α2+β2=(α+β)2−2αβ\alpha^2 + \beta^2 = (\alpha + \beta)^2 - 2\alpha\beta
  • α3+β3=(α+β)3−3αβ(α+β)\alpha^3 + \beta^3 = (\alpha + \beta)^3 - 3\alpha\beta(\alpha + \beta)
  • ∣α−β∣=(α+β)2−4αβ=D∣a∣|\alpha - \beta| = \sqrt{(\alpha + \beta)^2 - 4\alpha\beta} = \frac{\sqrt{D}}{|a|}
  • Common root condition: a1a2=b1b2=c1c2\text{Common root condition: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}
  • Biquadratic: ax4+bx2+c=0⇒substitute x2=t\text{Biquadratic: } ax^4 + bx^2 + c = 0 \Rightarrow \text{substitute } x^2 = t
  • Wavy curve method for f(x)>0 or f(x)<0\text{Wavy curve method for } f(x) > 0 \text{ or } f(x) < 0