A quadratic equation in x is an equation of the form ax2 + bx + c = 0, where a, b, c are real numbers and a is not 0. The word "quadratic" comes from "quadratus" (Latin for square). Quadratic equations arise in problems involving area, projectile motion, and many real-life situations.
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Key Concepts
Standard Form: ax2 + bx + c = 0, a not= 0. The left side is a quadratic polynomial.
Solution (Root): A value of x that satisfies the equation. A quadratic equation has at most two roots.
Methods of Solving:
1. Factorisation: Express ax2 + bx + c as a product of two linear factors. Set each factor to zero.
2. Completing the Square: Rewrite ax2 + bx + c = 0 as (x + p)2 = q, then take square roots.
3. Quadratic Formula: x = (-b ± √(b2 - 4ac)) / (2a)
- Discriminant (D): D = b2 - 4ac
- D > 0: Two distinct real roots
- D = 0: Two equal real roots (one repeated root)
- D < 0: No real roots (roots are imaginary)
Nature of Roots Summary:
| D | Nature |
|---|--------|
| D > 0 | Real and distinct |
| D = 0 | Real and equal |
| D < 0 | No real roots |
Key formulas
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Worked Examples
Solve x2 - 5x + 6 = 0 by factorisation.
We need two numbers whose product is 6 and sum is -5: -2 and -3.
x2 - 5x + 6 = (x - 2)(x - 3) = 0.
Roots: x = 2 and x = 3.
Solve 2x2 - 5x + 3 = 0 by factorisation.
Product = 2 x 3 = 6; we need numbers summing to -5: -2 and -3.
2x2 - 2x - 3x + 3 = 2x(x - 1) - 3(x - 1) = (2x - 3)(x - 1) = 0.
Roots: x = 3/2 and x = 1.
Solve x2 + 4x - 5 = 0 by completing the square.
x2 + 4x = 5.
Add (4/2)2 = 4 to both sides: x2 + 4x + 4 = 9.
(x + 2)2 = 9. x + 2 = ±3.
x = 1 or x = -5.
Solve 3x2 - 5x + 2 = 0 using the quadratic formula.
a = 3, b = -5, c = 2. D = 25 - 24 = 1.
x = (5 ± 1) / 6. x = 1 or x = 2/3.
Roots: x = 1 and x = 2/3.
Find the discriminant and nature of roots of x2 + x + 1 = 0.
D = 12 - 4(1)(1) = 1 - 4 = -3. D < 0. No real roots.
A train travels 360 km at a uniform speed. If the speed were 5 km/h more, it would take 1 hour less. Find the speed.
Let speed = x km/h. Time = 360/x hours.
New time = 360/(x+5) hours. Difference = 1 h:
360/x - 360/(x+5) = 1.
360(x+5) - 360x = x(x+5).
1800 = x2 + 5x.
x2 + 5x - 1800 = 0.
(x + 45)(x - 40) = 0. x = 40 (taking positive value).
Speed = 40 km/h.
For what value of k does kx2 + 2x + 1 = 0 have equal roots?
For equal roots, D = 0: 4 - 4k = 0, so k = 1.
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Key Formulas
Key formulas
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Common mistakes
- Forgetting that a quadratic equation must have a = not 0. If a = 0, it becomes linear.
- Taking only the positive square root in completing the square — always write ± .
- Using D to check nature of roots but then forgetting that D < 0 means no real roots, not "no roots."
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Summary
Quadratic equations are solved by factorisation, completing the square, or the formula. The discriminant is a quick indicator of the nature of roots without fully solving the equation. These skills are foundational for higher algebra and calculus.