Equations Reducible to Quadratic
Quadratic Equations
7
JEE Qs
8%
Hard
60
min
Always verify the obtained solutions by substituting them back into the original equation, especially for equations involving roots or fractions, to avoid extraneous roots and ensure validity.
🧮 Key Formulas
✅ Key Points for JEE
- 1The primary strategy is to identify a suitable common expression or term in the equation and substitute it with a new variable to transform the equation into a standard quadratic form (ay² + by + c = 0).
- 2Always verify the solutions obtained for the new variable in the context of the original equation's domain. For example, if y = sqrt(x), then y must be non-negative.
- 3After solving for the substituted variable, remember to back-substitute to find the solutions for the original variable.
- 4Be vigilant for extraneous roots, especially when squaring both sides of an equation, or when the original equation involves rational expressions (variables in the denominator) or radical expressions. Always substitute final solutions back into the original equation to check validity.
- 5Recognize common types of reducible equations: biquadratic equations (ax^4 + bx^2 + c = 0), reciprocal equations, equations involving (ax+b)/(cx+d) and similar terms, and equations involving radical signs.
⚠️ Common Mistakes
- ✕Failing to verify solutions in the original equation, which often leads to including extraneous roots, especially for rational or irrational equations.
- ✕Not considering the domain restrictions imposed by the substitution (e.g., if `t = sqrt(x)`, then `t` must be non-negative; if `t = x^2`, then `t` must be non-negative).
- ✕Algebraic errors during the substitution process or while solving the resulting quadratic equation.
- ✕Confusing the solutions for the substituted variable with the solutions for the original variable (e.g., reporting 'y' values instead of 'x' values).
- ✕Dividing by a variable expression without considering the case where that expression might be zero, potentially losing valid solutions.
📝 Practice Questions
See allQ22.The roots of the quadratic equation 3x2 −px + q = 0 are 10th and 11th terms of an arithmetic progression with common difference 32 . If the sum of the first 11 terms of this arithmetic progression is 88 , then q −2p is equal to -.
Q13. The number of real solution(s) of the equation x2 + 3x + 2 = min{|x −3|, |x + 2|} is : (1) 1 (2) 0 (3) 2 (4) 3
Q17.Let αθ and βθ be the distinct roots of 2x2 + (cos θ)x −1 = 0, θ ∈(0, 2π). If m and M are the minimum and the maximum values of α4θ + β4θ , then 16(M + m) equals : (1) 24 (2) 25 (3) 17 (4) 27
Q22.If the equation a(b −c)x2 + b(c −a)x + c(a −b) = 0 has equal roots, where a + c = 15 and b = 365 , then a2 + c2 is equal to
Q7. Let the line passing through the points (−1, 2, 1) and parallel to the line x−12 = y+13 = 4z intersect the line y−3 x+2 3 = 2 = z−41 at the point P . Then the distance of P from the point Q(4, −5, 1) is (1) 5 (2) 5√5 (3) 5√6 (4) 10
Q13.The sum, of the squares of all the roots of the equation x2 + |2x −3| −4 = 0, is (1) 3(3 −√2) (2) 6(3 −√2) (3) 6(2 −√2) (4) 3(2 −√2)
NCERT Chapters
- Class 11 Maths Ch 5: Quadratic Equations
- Class 10 Maths Ch 4: Quadratic Equations