Questions
5–8 questions per exam
Difficulty
Medium
Importance
Core — high scoring, never skip
Overview
Quadratic equations are second-degree polynomial equations of the form ax²+bx+c=0. Mastering these is crucial for board exams and competitive aptitude tests as they form the foundation for solving complex algebraic, geometric, and physics-related word problems.
Factorisation Method
This method involves splitting the middle term to express the quadratic as a product of two linear binomials. It is the most efficient technique when the roots are rational and easily identifiable.
- Standard form: ax²+bx+c=0 where a≠0
- Splitting the middle term: Find p and q such that p+q=b and pq=ac
- Setting factors (x-p)(x-q)=0 to zero
- Applicable only when discriminant is a perfect square
Quadratic Formula (Sridharacharya Formula)
The quadratic formula provides a universal method to solve any quadratic equation, regardless of whether it can be factored easily. It is the primary tool for finding roots when the equation is complex or features irrational values.
- Formula: x = [-b ± √(b² - 4ac)] / 2a
- Discriminant (D) = b² - 4ac
- Always calculate D first to simplify the process
- Valid for all real coefficients a, b, and c
Nature of Roots
The discriminant determines the nature and type of roots without needing to solve the entire equation. This is a frequently tested concept in objective exams.
- If D > 0: Two distinct real roots
- If D = 0: Two equal real roots
- If D < 0: No real roots (complex roots)
- D > 0 and perfect square: Rational roots
- D > 0 and not a perfect square: Irrational roots
Relationship Between Roots and Coefficients
Vieta's formulas provide a direct link between the roots and the coefficients of the polynomial, allowing for quick construction of equations or calculation of root expressions.
- Sum of roots (α+β) = -b/a
- Product of roots (αβ) = c/a
- Formation of quadratic: x² - (sum of roots)x + (product of roots) = 0
- Difference of roots = |√D / a|
Formula Sheet
ax² + bx + c = 0
x = (-b ± √(b² - 4ac)) / 2a
D = b² - 4ac
α + β = -b/a
αβ = c/a
x² - (α+β)x + αβ = 0
Exam Tip
Always verify your calculated roots by checking if their sum equals -b/a and their product equals c/a; it takes five seconds and prevents silly errors.
Common Mistakes
- Forgetting the negative sign in the sum of roots formula (-b/a).
- Failing to equate the quadratic to zero before applying the formula.
- Incorrectly identifying 'a', 'b', and 'c' when the equation is not in standard order.
More Revision Notes
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