Key Concepts
- 1Quadratic Formula
- 2Discriminant & Nature of Roots
- 3Sum of roots
- 4Product of roots
- 5Factorisation method
- 6Completing the square
- 7Equal roots condition
- 8Quadratic Formula
- 9Discriminant & Nature of Roots
- 10Sum of roots
Important Formulas & Facts
x = [-b ± √(b² - 4ac)] / 2a
D = b² - 4ac. D > 0: two distinct real. D = 0: two equal. D < 0: no real roots.
For ax² + bx + c = 0: sum = -b/a
For ax² + bx + c = 0: product = c/a
Split the middle term such that product = ac and sum = b.
Add (b/2a)² to both sides to make a perfect square on LHS.
D = 0, i.e., b² = 4ac
x = [-b ± √(b² - 4ac)] / 2a
D = b² - 4ac. D > 0: two distinct real. D = 0: two equal. D < 0: no real roots.
For ax² + bx + c = 0: sum = -b/a
Must-Know Questions
Q1Find the roots of x² - 7x + 10 = 0.
x² - 7x + 10 = (x-5)(x-2) = 0. x = 5 or x = 2.
Q2The discriminant of 2x² - 5x + 3 = 0 is:
D = b² - 4ac = (-5)² - 4(2)(3) = 25 - 24 = 1.
Q3For what value of k does kx² + 6x + 1 = 0 have equal roots?
Equal roots: D = 0. 36 - 4k = 0 → k = 9.
Q4Solve: 2x² + x - 6 = 0 using the quadratic formula.
x = [-b ± √(b²-4ac)] / 2a = [-1 ± √(1+48)] / 4 = [-1 ± 7] / 4. x = 6/4 = 3/2 or x = -8/4 = -2.
Q5If x² + 4x + k = 0 has real and distinct roots, then:
Real distinct: D > 0. 16 - 4k > 0 → k < 4.
Practice Quadratic Equations
Reinforce what you just revised with practice questions