Vieta's Formulas — Sum and product of roots
Quadratic Equations
7
JEE Qs
8%
Hard
60
min
Mastering Vieta's formulas simplifies problems involving roots without needing to explicitly find them, especially for symmetric expressions and finding relationships between coefficients and root properties.
🧮 Key Formulas
✅ Key Points for JEE
- 1Vieta's formulas establish a direct relationship between the coefficients of a quadratic polynomial and the sum/product of its roots, eliminating the need to solve for roots explicitly.
- 2They are crucial for forming a quadratic equation when the sum and product of its roots are known.
- 3Highly effective for simplifying symmetric expressions involving roots (e.g., α^2 + β^2, 1/α + 1/β) by replacing them with expressions of α+β and αβ.
- 4Conditions on roots (e.g., one root is reciprocal of other, roots are equal in magnitude but opposite in sign) can be directly translated into conditions on coefficients using these formulas.
⚠️ Common Mistakes
- ✕Incorrectly identifying coefficients 'a', 'b', 'c' in equations not in standard ax^2 + bx + c = 0 form (e.g., when the variable is not x, or terms are not ordered).
- ✕Sign errors, especially with the sum of roots formula (e.g., using b/a instead of -b/a).
- ✕Forgetting to divide by 'a' (the coefficient of x^2) when calculating the sum and product of roots.
- ✕Errors in algebraic manipulation when substituting α+β and αβ into complex expressions involving roots.
📝 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 10 Mathematics Ch 4: Quadratic Equations
- Class 11 Mathematics Ch 5: Quadratic Equations