Quadratic Equations
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision.
Quadratic Equations — Key Facts for CUET • Discriminant Formula: For ax² + bx + c = 0, discriminant D = b² - 4ac determines nature of roots • Most tested CUET concept: Relationship between coefficients and roots (α + β = -b/a, αβ = c/a) • Common mistake students make: Forgetting that ‘a’ cannot be zero; confusing signs when applying the quadratic formula • Key technique or method to solve quickly: Use factorization when numbers are nice; use Shridharacharya formula when factorization is difficult • Important exception or special case: When b = 0, equation becomes ax² + c = 0; roots are ±√(-c/a) (imaginary if c/a > 0) • Most frequent question type in CUET: Forming quadratic equation from given roots, and word problems involving area/perimeter ⚡ Exam tip: In multiple choice questions, if options are numbers, substitute the options into the equation to quickly verify which satisfies it — saves time during CUET
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Quadratic Equations — CUET Study Guide
A quadratic equation in one variable is of the standard form ax² + bx + c = 0, where a ≠ 0. The roots of this equation are given by the quadratic formula: x = (-b ± √(b² - 4ac)) / 2a. The expression D = b² - 4ac is called the discriminant and determines the nature of roots. If D > 0, roots are real and unequal; if D = 0, roots are real and equal; if D < 0, roots are complex conjugates.
The Sum and Product of roots (Vieta’s formulas) are fundamental: α + β = -b/a and αβ = c/a. These allow quick formation of equations when roots are known: x² - (sum)x + (product) = 0. Common traps include misapplying signs and forgetting the ‘a’ coefficient in the denominator.
Solved Examples:
-
Find roots of x² - 5x + 6 = 0 → Factor: (x-2)(x-3) = 0, so x = 2, 3. Using formula: D = 25-24 = 1, roots = (5±1)/2 = 2, 3.
-
If roots are 3 and -2, form equation → Sum = 1, Product = -6. Equation: x² - x - 6 = 0.
Word problems typically involve consecutive integers, area calculations, or speed-distance-time relationships leading to quadratic setups.
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer timeline.
Quadratic Equations — Comprehensive CUET Notes
Theory and Proofs: A quadratic equation ax² + bx + c = 0 (a ≠ 0) represents a parabola when graphed. The vertex form is a(x - h)² + k, where h = -b/2a gives the axis of symmetry. The Maximum or Minimum value occurs at x = -b/2a, with value = -D/4a. For real roots, the parabola must intersect the x-axis, requiring D ≥ 0.
Relationship Between Coefficients and Roots: Given roots α, β, we derive: α + β = -b/a and αβ = c/a by expanding a(x - α)(x - β) = ax² - a(α+β)x + aαβ. When roots are given as α, β, the quadratic is: ax² - a(α+β)x + aα Content adapted based on your selected roadmap duration. Switch tiers using the pill selector above.