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📑 In This Chapter Guide (Table of Contents)
1. Standard Form & Methods of Solving Quadratic Equations
A quadratic equation in variable x is an equation of the form:
ax² + bx + c = 0 (where a, b, c are real numbers and a ≠ 0)
Method 1: Factorization by Splitting the Middle Term:
Find two numbers p and q such that their sum p + q = b and their product p · q = a · c. Express the middle term as px + qx, group the terms, and factor out binomial roots.
Method 2: Quadratic Formula (Shreedharacharya’s Rule):
x = [ -b ± √(b² - 4ac) ] / (2a)
2. Nature of Roots & The Discriminant (D)
The expression D = b² - 4ac is called the Discriminant because it discriminates the character of the solutions:
| Value of Discriminant D | Nature of Roots | Roots Formula |
|---|---|---|
D > 0 (Positive) |
Two distinct real roots | x = (-b + √D)/2a and x = (-b - √D)/2a |
D = 0 (Zero) |
Two equal real roots (coincident) | x = -b / (2a) (repeated twice) |
D < 0 (Negative) |
No real roots (Imaginary roots) | No real solutions exist in ℝ |
3. Solving Classic Train Speed & Work-Time Word Problems
Standard Train Speed Problem Template:
"A train travels 360 km at a uniform speed. If the speed had been 5 km/h more, it would have taken 1 hour less for the same journey. Find the original speed."
1. Let original speed = x km/h. Increased speed = (x + 5) km/h.
2. Time taken at original speed: t₁ = 360 / x hours.
3. Time taken at increased speed: t₂ = 360 / (x + 5) hours.
4. Equation: t₁ - t₂ = 1 ⟹ 360/x - 360/(x + 5) = 1
5. Simplify: 360[(x + 5 - x) / (x(x + 5))] = 1 ⟹ 1800 = x² + 5x
6. x² + 5x - 1800 = 0 ⟹ (x + 45)(x - 40) = 0
7. Since speed cannot be negative, x = 40 km/h. The original speed is 40 km/h.
💡 Frequently Asked Questions (FAQ)
❓ Find the values of k for which 2x² + kx + 3 = 0 has two equal real roots.
For equal real roots, the discriminant must be zero: D = b² - 4ac = 0. Here a = 2, b = k, c = 3. Thus: k² - 4(2)(3) = 0 ⟹ k² - 24 = 0 ⟹ k² = 24 ⟹ k = ±√24 = ±2√6.
❓ Can the sum of ages of two friends be 20 years, with product of their ages 4 years ago being 48?
Let friend A age = x. Friend B age = 20 - x. Four years ago: (x - 4)(16 - x) = 48 ⟹ 16x - x² - 64 + 4x = 48 ⟹ x² - 20x + 112 = 0. Discriminant D = (-20)² - 4(1)(112) = 400 - 448 = -48 < 0. Since D < 0, no real solution exists. This situation is mathematically impossible.
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