Quadratic Equations in One Variable
Introduction
A quadratic equation in one variable x is an equation of the form ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0. In ICSE Class 10, you learn multiple methods to solve quadratic equations and analyse the nature of roots using the discriminant.
Standard Form
ax² + bx + c = 0, where a ≠ 0
The highest power of the variable is 2, hence the name 'quadratic' (quad = square).
Methods of Solving Quadratic Equations
Method 1: Factorisation (Splitting the Middle Term)
Step 1: Write the equation in standard form. Step 2: Find two numbers whose product = ac and sum = b. Step 3: Split the middle term using these numbers. Step 4: Factor by grouping. Step 5: Set each factor to zero and solve.
Example: Solve x² − 5x + 6 = 0
- Find two numbers: product = 6, sum = −5 → numbers are −2 and −3
- x² − 2x − 3x + 6 = 0
- x(x − 2) − 3(x − 2) = 0
- (x − 2)(x − 3) = 0
- x = 2 or x = 3
Method 2: Completing the Square
Step 1: Divide by a (coefficient of x²). Step 2: Move c/a to the RHS. Step 3: Add (b/2a)² to both sides. Step 4: Write LHS as a perfect square. Step 5: Take square root and solve.
Example: Solve x² + 6x − 7 = 0
- x² + 6x = 7
- Add (6/2)² = 9 to both sides: x² + 6x + 9 = 16
- (x + 3)² = 16
- x + 3 = ±4
- x = 1 or x = −7
Method 3: Quadratic Formula
x = [−b ± √(b² − 4ac)] / 2a
The expression D = b² − 4ac is called the discriminant.
Example: Solve 2x² − 4x − 3 = 0
- a = 2, b = −4, c = −3
- D = (−4)² − 4(2)(−3) = 16 + 24 = 40
- x = [4 ± √40] / 4 = [4 ± 2√10] / 4
- x = (2 ± √10) / 2
Nature of Roots (Discriminant Analysis)
| Discriminant (D) | Nature of Roots | Example |
|---|---|---|
| D > 0 and perfect square | Real, rational, distinct | x² − 5x + 6 = 0 |
| D > 0 and not a perfect square | Real, irrational, distinct | x² − 4x + 2 = 0 |
| D = 0 | Real, rational, equal (coincident) | x² − 6x + 9 = 0 |
| D < 0 | Imaginary (no real roots) | x² + x + 1 = 0 |
Word Problems — Step-by-Step Approach
Example 1: Number Problem
The sum of the squares of two consecutive positive integers is 365. Find the integers.
Solution: Let the integers be x and x + 1. x² + (x + 1)² = 365 x² + x² + 2x + 1 = 365 2x² + 2x − 364 = 0 x² + x − 182 = 0
Factorising: (x + 14)(x − 13) = 0 x = 13 or x = −14 (reject, since positive)
The integers are 13 and 14.
Example 2: Geometry Problem
A rectangular field has length 10 m more than its width. Its area is 600 m². Find the dimensions.
Solution: Let width = x m, length = (x + 10) m. x(x + 10) = 600 x² + 10x − 600 = 0
Using formula: D = 100 + 2400 = 2500 x = [−10 ± 50] / 2 = 20 or −30 (reject negative)
Width = 20 m, Length = 30 m
Example 3: Age Problem
Five years ago, the product of ages of a father and son was 180. The sum of their present ages is 50. Find their present ages.
Solution: Let father's age = x, son's age = 50 − x. Five years ago: (x − 5)(45 − x) = 180 45x − x² − 225 + 5x = 180 −x² + 50x − 405 = 0 x² − 50x + 405 = 0
D = 2500 − 1620 = 880 x = [50 ± √880] / 2 = [50 ± 4√55] / 2 = 25 ± 2√55
x ≈ 39.83 or 10.17 Father's age ≈ 40 years, Son's age ≈ 10 years
Common Mistakes and Fixes
| Mistake | Fix |
|---|---|
| Forgetting to write in standard form | Always bring all terms to LHS before solving |
| Incorrect factorisation | Verify by expanding the factors back |
| Sign errors in quadratic formula | Write x = [−b ± √(b² − 4ac)] / 2a carefully |
| Rejecting negative solutions without checking context | Only reject if the problem implies positive values |
| Forgetting to include ± when taking square root | Remember: √(k²) = ±k |
ICSE Exam Focus
Quadratic equations carry 10–14 marks in ICSE exams — one of the most important topics. Questions include:
- Solving using factorisation.
- Solving using the formula.
- Nature of roots problems.
- Word problems (numbers, ages, geometry, speed-distance-time, money).
- Finding unknown coefficients given conditions on roots.
Marks Blueprint:
| Topic | Marks |
|---|---|
| Solving quadratic (factorisation) | 3 |
| Solving quadratic (formula/completing square) | 3 |
| Nature of roots / discriminant | 2 |
| Word problem | 4–6 |
Self-Test Questions
-
Solve 3x² − 11x + 6 = 0 by factorisation.
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Solve 2x² + 5x − 3 = 0 using the quadratic formula.
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For what value of k does the equation kx² − 8x + 4 = 0 have equal roots?
-
The speed of a boat in still water is 8 km/h. It takes 6 hours to travel 18 km upstream and 18 km downstream. Find the speed of the stream.
-
A two-digit number is such that the product of its digits is 18. When 63 is subtracted from the number, the digits are reversed. Find the number.
-
Prove that the quadratic equation x² + 2x + 5 = 0 has no real roots.
In ICSE exams, show every step clearly — partial credit is generous for correct algebraic manipulation even if the final answer has a computational slip.
