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Quadratic Equations

CRPF Constable · Quantitative Aptitude

Also prepare for:BSF Const CISF Const ITBP Const UP Const

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Core Concept

Read this first — the foundation of the topic

🔑Core Properties and Rules

Every quadratic equation has exactly two roots (solutions). These roots can be real and equal, real and unequal, or imaginary. The nature of roots depends on the discriminant (b² - 4ac). When discriminant > 0, roots are real and unequal.

When discriminant = 0, roots are real and equal. When discriminant < 0, roots are imaginary

💡Key Formulas

Quadratic Formula: x = [-b ± √(b² - 4ac)] / 2a Sum of roots = -b/a Product of roots = c/a If roots are α and β, then equation is: x² - (α + β)x + αβ = 0

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Exam Patterns

What examiners ask — read before attempting PYQs

✏️Worked Example 1

1

Identify a = 2, b = -7, c = 3

2

Find ac = 2 × 3 = 6

3

Find two numbers whose product = 6 and sum = -7 These are -6 and -1 (since -6 × -1 = 6 and -6 + (-1) = -7)

4

Rewrite: 2x² - 6x - x + 3 = 0

5

Factor: 2x(x - 3) - 1(x - 3) = 0

6

(2x - 1)(x - 3) = 0

7

Roots are x = 1/2 and x = 3 Worked Example 2: Find the equation whose roots are 3 and -2.

1

Sum of roots = 3 + (-2) = 1

2

Product of roots = 3 × (-2) = -6

3

Using x² - (sum of roots)x + (product of roots) = 0

4

Required equation: x² - x - 6 = 0 Shortcut Trick #3 - Vieta's Relations: Instead of finding individual roots, use sum and product formulas directly. This is especially useful when questions ask about expressions involving roots without finding the actual roots. Common Trap - The #1 Mistake: Students often forget to check if 'a' equals zero. If a = 0, the equation becomes linear (bx + c = 0), not quadratic. Always verify that the coefficient of x² is non-zero before applying quadratic methods. Another common error is sign mistakes while applying the quadratic formula, especially with the discriminant calculation

📋Exam Strategy

For SSC CGL, master middle term splitting first as it's faster than the quadratic formula. Practice identifying perfect squares quickly. Word problems often lead to quadratic equations, so focus on translation skills. Time management is crucial - spend maximum 2 minutes per quadratic equation question.

🔑 Key Points

Standard form: ax² + bx + c = 0 where a ≠ 0
Discriminant = b² - 4ac determines nature of roots
Sum of roots = -b/a, Product of roots = c/a
Middle term splitting is faster than quadratic formula for most SSC questions
If b² = 4ac, equation is perfect square with root = -b/2a
Every quadratic equation has exactly two roots
Roots can be real unequal, real equal, or imaginary
For forming equation from roots: x² - (sum)x + (product) = 0
Use Vieta's relations when questions involve expressions of roots
Always verify coefficient of x² is non-zero before applying quadratic methods

📌 Exam Facts

Discriminant > 0 means real and unequal roots
Discriminant = 0 means real and equal roots
Discriminant < 0 means imaginary roots
Maximum value of quadratic expression ax² + bx + c occurs at x = -b/2a when a < 0
Minimum value of quadratic expression ax² + bx + c occurs at x = -b/2a when a > 0
If roots are reciprocals of each other, then a = c
If one root is negative of the other, then b = 0
Quadratic formula: x = [-b ± √(b² - 4ac)] / 2a

🚀 60-Second Revision

Formula: Sum = -b/a, Product = c/a, Discriminant = b² - 4ac
Remember: Use middle term splitting for faster solutions in SSC
Trap: Always check if coefficient of x² is non-zero
Shortcut: If b² = 4ac, root is -b/2a (perfect square)
Pattern: 2-3 questions appear in every SSC CGL paper
Strategy: Master Vieta's relations for expressions involving roots
Time limit: Maximum 2 minutes per quadratic equation question

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