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Chapter Analysis
Intermediate11 pages • EnglishQuick Summary
In this chapter, the concept of quadratic equations is explored in detail. The chapter examines the standard form of a quadratic equation and explains how such equations can be solved using factorization, the quadratic formula, and completing the square method. Additionally, the chapter discusses the discriminant's role in determining the nature of roots and demonstrates applications of quadratic equations in real-life scenarios. Examples of quadratic equations in various contexts are also provided.
Key Topics
- •Standard form of quadratic equation
- •Methods of solving quadratic equations
- •Nature and types of roots
- •Applications of quadratic equations
- •Factorization
- •Quadratic formula
- •Completing the Square
- •Discriminant and its significance
Learning Objectives
- ✓Understand the standard form of a quadratic equation.
- ✓Solve quadratic equations using different methods such as factorization and quadratic formula.
- ✓Identify the nature of roots using the discriminant.
- ✓Apply quadratic equations to solve real-life mathematical problems.
- ✓Develop a conceptual understanding of quadratic properties.
- ✓Analyze and interpret solutions of quadratic equations in practical contexts.
Questions in Chapter
Find the nature of the roots of the following quadratic equations. If the real roots exist, find them: (i) 2x^2 – 3x + 5 = 0 (ii) 3x^2 – 4√3 x + 4 = 0 (iii) 2x^2 – 6x + 3 = 0
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Find the values of k for each of the following quadratic equations, so that they have two equal roots. (i) 2x^2 + kx + 3 = 0 (ii) kx (x – 2) + 6 = 0
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Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m^2? If so, find its length and breadth.
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Is the following situation possible? If so, determine their present ages. The sum of the ages of two friends is 20 years. Four years ago, the product of their ages in years was 48.
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Is it possible to design a rectangular park of perimeter 80 m and area 400 m^2? If so, find its length and breadth.
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Additional Practice Questions
Solve the quadratic equation: x^2 - 4x + 4 = 0 by factoring.
easyAnswer: The equation can be factored as (x-2)(x-2) = 0, giving a repeated root of x = 2.
Find the roots of the equation 3x^2 - 12x + 12 = 0 using the quadratic formula.
mediumAnswer: Using the quadratic formula, x = [12 ± √(144 - 144)]/6, the roots are x = 2 with multiplicity 2.
A rectangle has its length 3 meters more than twice its width. If the area of the rectangle is 50 square meters, find the width.
mediumAnswer: Let W be the width, then length is 2W + 3. Area equation W(2W + 3) = 50 gives a quadratic 2W^2 + 3W - 50 = 0. Solving this, W ≈ 4.2 m.
Find two consecutive integers such that the square of the smaller is 4 less than 10 times the larger one.
mediumAnswer: Let the integers be x and x+1. Then, x^2 = 10(x + 1) - 4. Simplifying, x^2 - 10x - 14 = 0. Solving gives x = 12, -2, so integers are 12 and 13.
A train travels a distance of 360 km at a uniform speed. If the speed is reduced by 5 km/h, the train takes 2 hours more. Find the original speed.
hardAnswer: Let the original speed be x km/h. Thus, 360/x - 360/(x-5) = 2. Solving the quadratic equation derived gives x ≈ 60 km/h.