Quadratic Equations is a chapter in the NCERT Maths textbook for Class 10 that deals with polynomial equations of degree two.
Key Concepts :
Quadratic Equation: An equation of the form ax² + bx + c = 0, where a, b, and c are real numbers and a ≠ 0.
Roots of a Quadratic Equation: The values of x that satisfy the quadratic equation.
Methods of Solving Quadratic Equations:
Factorization
Completing the Square
Quadratic Formula
Discriminant (D): The value of b² – 4ac, used to determine the nature of roots.
If D > 0, two distinct real roots
If D < 0, no real roots
Nature of Roots: Based on the discriminant, we can determine if the roots are real, equal, or imaginary.
Important Points
Quadratic equations have applications in various real-life situations, such as physics, engineering, and economics.
In essence, this chapter provides a comprehensive understanding of quadratic equations, their properties, and different methods to solve them. It also introduces the concept of the discriminant, which helps in determining the nature of roots without actually solving the equation.
Exercise 4.1
1. Check whether the following are quadratic equations:
(i) (x+ 1)2=2(x-3)
(ii) x – 2x = (- 2) (3-x)
(iii) (x – 2) (x + 1) = (x – 1) (x + 3)
(iv) (x – 3) (2x + 1) = x (x + 5)
(v) (2x – 1) (x – 3) = (x + 5) (x – 1)
(vi) x2 + 3x + 1 = (x – 2)2
(vii) (x + 2)3 = 2x(x2 – 1)
(viii) x3 -4x2 -x + 1 = (x-2)3
Ans :
2. Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is 528 m2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan’s mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan’s present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Ans :
(i) Rectangular Plot
Then, the length of the plot is (2x + 1) meters.
Area of the plot = length * breadth 528 = x(2x + 1)
2x² + x – 528 = 0
This is a quadratic equation.
(ii) Consecutive Positive Integers
Let the smaller integer be x.
Then, the next consecutive integer is x + 1.
Product of the integers = x(x + 1) = 306
x² + x – 306 = 0
This is a quadratic equation.
(iii) Rohan’s Age
Let Rohan’s present age be x years.
His mother’s present age = (x + 26) years.
After 3 years, Rohan’s age will be (x + 3) years and his mother’s age will be (x + 29) years.
Product of their ages after 3 years = (x + 3)(x + 29) = 360
x² + 32x + 87 = 360
x² + 32x – 273 = 0
This is a quadratic equation.
Exercise 4.2
1. Find the roots of the following quadratic equations by factorisation:
(i) x2 -3x – 10 = 0
(ii) 2x2 + x – 6 = 0
(iii) √2x2 + 7x + 5√2 = 0
(iv) 2x2 – x + 1/8= 0 8
(v) 100 x2 – 20 X + 1 = 0
Ans :
(i)
We need to find two numbers whose product is -10 and whose sum is -3.
These numbers are -5 and 2.
So, we can factorize the equation as:
(x – 5)(x + 2) = 0
Therefore, x – 5 = 0 or x + 2 = 0
x = 5 and x = -2.
(ii) 2x² + x – 6 = 0
We need to find two numbers whose product is -12 (2 * -6) and whose sum is 1.
These numbers are 4 and -3.
So, we can rewrite the equation as:
2x² + 4x – 3x – 6 = 0
Factor by grouping:
2x(x + 2) – 3(x + 2) = 0
(2x – 3)(x + 2) = 0
Therefore, 2x – 3 = 0 or x + 2 = 0
Roots are x = 3/2 and x = -2.
(iii) √2x² + 7x + 5√2 = 0
We need to find two numbers whose product is 10 (√2 * 5√2) and whose sum is 7.
These numbers are 5√2 and 2√2.
So, we can factorize the equation as:
(√2x + 5)(x + √2) = 0
Therefore, √2x + 5 = 0 or x + √2 = 0
Roots are x = -5/√2 and x = -√2.
(iv) 2x² – x + 1/8 = 0
16x² – 8x + 1 = 0
These numbers are -4 and -4.
So, we can factorize the equation as:
(4x – 1)² = 0
Therefore, 4x – 1 = 0
Root is x = 1/4.
(v) 100x² – 20x + 1 = 0
This is a perfect square trinomial:
(10x – 1)² = 0
Therefore, 10x – 1 = 0
Root is x = 1/10.
These are the roots of the given quadratic equations found by factorization.
3. Find two numbers whose sum is 27 and product is 182.
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4. Find two consecutive positive integers, the sum of whose squares is 365.
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5. The altitude of a right triangle is 7 cm less than its base. If the hypotenuse is 13 cm, find the other two sides.
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6. A cottage industry produces a certain number of pottery articles in a day. It was observed on a particular day that the cost of production of each article (in rupees) was 3 more than twice the number of articles produced on that day. If the total cost of production on that day was ₹90, find the number of articles produced and the cost of each article.
Ans :
The cost of production of each article is 2x + 3 rupees.
Total cost of production = Number of articles * Cost per article Therefore, x(2x + 3) = 90
Expanding the equation: 2x² + 3x = 90 2x² + 3x – 90 = 0
Factorize the quadratic equation:
(2x + 15)(x – 6) = 0
So, x = -15/2 or x = 6.
Since the number of articles cannot be negative, x = 6.
Therefore, the number of articles produced is 6. The cost of each article = 2x + 3 = 2(6) + 3 = 15 rupees.
Exercise 4.3
1. Find the nature of the roots of the following quadratic equations. If the real roots exist, find them:
(i) 2x² -3x + 5 = 0
(ii) 3x2 – 4√3x + 4 = 0
(iii) 2x2-6x + 3 = 0
Ans :
For a quadratic equation of the form ax² + bx + c = 0, the discriminant (D) is given by:
- D = b² – 4ac
- If D > 0, the equation has two distinct real roots.
- If D < 0, the equation has no real roots (complex roots).
Solving the Equations
(i) 2x² – 3x + 5 = 0
- a = 2, b = -3, c = 5
- D = (-3)² – 4(2)(5) = 9 – 40 = -31 Since D < 0,
- the equation has no real roots.
(ii) 3x² – 4√3x + 4 = 0
- a = 3,
- b = -4√3,
- c = 4
- D = (-4√3)² – 4(3)(4) = 48 – 48 = 0 Since D = 0,
- the equation has two equal real roots. To find the root, we can use the formula x = -b / 2a:
- x = -(-4√3) / (2 * 3) = 2√3 / 3
(iii) 2x² – 6x + 3 = 0
- a = 2, b = -6, c = 3
- D = (-6)² – 4(2)(3) = 36 – 24 = 12 Since D > 0, the equation has two distinct real roots. Using the quadratic formula:
- x = (-b ± √D) / (2a) = (6 ± √12) / (2 * 2) = (6 ± 2√3) / 4 = (3 ± √3) / 2
Therefore, the roots are (3 + √3)/2 and (3 – √3)/2.
2. Find the values of k for each of the following quadratic equations, so that they have two equal roots.
(1) 2x2 + kx + 3 = 0
(2) kx (x – 2) + 6 = 0
Ans :
(1) 2x² + kx + 3 = 0
Here, a = 2, b = k, and c = 3.
Since the equation has equal roots, D = 0.
Therefore, k² – 4(2)(3) = 0
k² – 24 = 0
k² = 24
k = ±√24
k = ± 2√6
(2) kx(x – 2) + 6 = 0
Expanding the equation, we get:
kx² – 2kx + 6 = 0
Here, a = k, b = -2k, and c = 6.
Since the equation has equal roots, D = 0.
Therefore, (-2k)² – 4(k)(6) = 0
4k² – 24k = 0
4k(k – 6) = 0
So, either 4k = 0 or
k – 6 = 0
Hence, k = 0 or
k = 6.
But if k = 0, the equation becomes 6 = 0, which is not possible.
Therefore, the only possible value for k is k = 6.
3. Is it possible to design a rectangular mango grove whose length is twice its breadth, and the area is 800 m2? If so, find its length and breadth.
Ans :
Let the breadth of the grove be x meters.
So, the length will be 2x meters.
Area of a rectangle
= length * breadth
Therefore, 2x * x = 800
2x² = 800
x² = 400
x = ± 20
Since length cannot be negative, we discard x = -20.
So, the breadth (x) = 20 meters.
And the length (2x) = 40 meters.
Yes, it is possible to design such a mango grove.The length of the grove is 40 meters and the breadth is 20 meters.
4. 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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5. Is it possible to design a rectangular park of perimeter 80 m and area 400 m2? If so, find its length and breadth.
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