Algebra: Complete Mastery

1. Linear Equations (2 Variables)

For equations \(a_1x + b_1y + c_1 = 0\) and \(a_2x + b_2y + c_2 = 0\):

• Unique Solution: \( \frac{a_1}{a_2} \neq \frac{b_1}{b_2} \)
• No Solution: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2} \)
• Infinitely Many Solutions: \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

2. Quadratic Equations & Nature of Roots

General Form: \( ax^2 + bx + c = 0 \). Discriminant \( D = b^2 - 4ac \).

• \( D > 0 \): Real & Distinct Roots (If D is a perfect square, roots are rational).
• \( D = 0 \): Real & Equal Roots.
• \( D < 0 \): Imaginary/Complex Roots (No real roots).

Sum & Product of Roots: \( \alpha + \beta = \frac{-b}{a} \) and \( \alpha\beta = \frac{c}{a} \).

Difference of Roots: \( (\alpha - \beta)^2 = (\alpha + \beta)^2 - 4\alpha\beta \).

3. Inequalities (x vs y) & RCE

To compare \(x\) and \(y\), solve both equations and compare their roots.

4. Advanced Mains Concepts

• Transformed Roots: If roots are \(\alpha, \beta\), the equation with roots \(\alpha^2, \beta^2\) is formed using Sum = \(\alpha^2+\beta^2\) and Product = \((\alpha\beta)^2\).
• Wrong Roots Trick: If someone uses wrong '\(p\)' and gets roots (a, b), their product is correct (\(q=ab\)). If they use wrong '\(q\)' and gets roots (c, d), their sum is correct (\(p=-(c+d)\)).
• Special Equations:
  • If \(x\) is common in all terms (e.g., \(x^3 - 17x^2 + 72x = 0\)), one root is \(x=0\).
  • For square root equations like \(3x - 39\sqrt{x} + 108 = 0\), substitute \(\sqrt{x} = a\).