Equations Reducible to Quadratic and Common Roots

Equations Reducible to Quadratic and Common Roots

This lesson covers the following key concepts:

• Biquadratic equations: ax⁴ + bx² + c = 0
• Reciprocal equations
• Equations solvable by substitution
• Common roots of two quadratic equations
• Condition for exactly one common root
• Condition for both roots common
• Applications to JEE problems

Important Formulas

• $$\text{Biquadratic: } ax^4 + bx^2 + c = 0 \Rightarrow \text{let } y = x^2$$
• $$\text{One common root: } (a_1c_2 - a_2c_1)^2 = (b_1c_2 - b_2c_1)(a_1b_2 - a_2b_1)$$
• $$\text{Both roots common: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$$
• $$\text{Reciprocal eq: } ax^4 + bx^3 + cx^2 + bx + a = 0 \Rightarrow \text{divide by } x^2$$
• $$\text{Let } y = x + \frac{1}{x} \text{ for reciprocal equations}$$