Quadratic Equations in one variable

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Class 10 Maths Chapter 5
Quadratic Equations in one variable
Important Questions

Q: Q1. What is a quadratic equation in one variable?

Ans: A quadratic equation in one variable is a polynomial equation of the form ax^2 +bx+c=0, \space where a, b, and c are constants, and x is the variable.

Q: Q2. How do you determine the roots of a quadratic equation?

Ans: The roots of a quadratic equation are the values of x that satisfy the equation and make it equal to zero. You can find the roots by factoring, using the quadratic formula, or completing the square.

Q: Q3. Can a quadratic equation have no real roots?

Ans: Yes, a quadratic equation can have no real roots if its discriminant is negative. In such cases, it has two complex roots.

Q: Q4. What does it mean if the coefficient a is positive in a quadratic equation?

Ans: If the coefficient a is positive, the parabola opens upward, and the quadratic equation has a minimum value. The graph is U-shaped.

Q: Q5. Are there any shortcuts to factorize quadratic equations quickly?

Ans: Yes, some common factorization patterns, such as the difference of squares or perfect squares, can be used as shortcuts to factorize quadratic equations efficiently.

Quadratic equations are a fundamental topic in algebra that deals with equations of the second degree, where the highest power of the variable is squared. In the context of ICSE Class 10 mathematics, quadratic equations in one variable are expressions of the form:

ax^2 + bx + c = 0

Where:

a, b, and c are constants, with a ≠ 0
x is the variable we’re solving for.
$ax^2$ represents the quadratic term.
bx represents the linear term.
c is the constant term.
The equation is set equal to zero. These quadratic equation class 10 ICSE important questions serve as invaluable practice tools for upcoming class tests, unit assessments, and examinations. They are strategically designed to bolster students’ confidence in their mathematical prowess and address any uncertainties or challenges they may encounter while studying this topic.

Table of Contents

Introduction

Understanding quadratic equation class 10 ICSE is essential for building a strong foundation in algebra and problem-solving. It equips students with the tools to analyse and solve a wide range of mathematical and practical problems. In your Class 10 quadratic equation in one variable, you will explore different methods to solve quadratic equations, understand the significance of the discriminant, and apply these concepts to real-life situations. This knowledge not only helps in achieving success in mathematics but also develops critical thinking and analytical skills that are valuable in many aspects of life. Key Concepts: Solving Quadratic Equations: Solving a quadratic equation means finding the values of x that make the equation true. Various methods are available for solving quadratic equations, including factoring, using the quadratic formula, and completing the square. Roots of a Quadratic Equation: The solutions to a quadratic equation (values of x that satisfy the equation) are called its roots or solutions. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots (if the discriminant is negative). Discriminant: The discriminant (Δ) is a crucial part of the quadratic formula and determines the nature of the roots: Δ>0: Two distinct real roots. Δ=0: One real root (repeated). Δ<0: Two complex roots. Applications: Quadratic equations are widely used to model various real-world scenarios, including physics (projectile motion), engineering (optimization), finance (interest calculations), and more.

What are Quadratic Equations in one variable?

Quadratic equations in one variable as we study in quadratic equation class 10 ICSE are mathematical expressions of the form:

// ax^2 + bx + c = 0

Where:

a, b, and c are constants, with a ≠ 0
x is the variable we’re solving for.
$ax^2$ represents the quadratic term.
bx represents the linear term.
c is the constant term.
The equation is set equal to zero.

Key concepts related to quadratic equations in one variable include: Solving Quadratic Equations: Solving a quadratic equation means finding the values of x that satisfy the equation, making it true. Various methods can be used to solve quadratic equations, including factoring, using the quadratic formula, and completing the square. Roots of a Quadratic Equation: The solutions to a quadratic equation (values of x that satisfy the equation) are called its roots or solutions. A quadratic equation can have two real roots, one real root (a repeated root), or two complex roots (if the discriminant is negative). Discriminant: The discriminant (Δ) is a crucial part of the quadratic formula and determines the nature of the roots: Δ>0: Two distinct real roots. Δ=0: One real root (repeated). Δ<0: Two complex roots. Applications: Quadratic equations are widely used to model various real-world scenarios, including physics (projectile motion), engineering (optimization), finance (interest calculations), and more.

Class 10 Quadratic Equations in one variable Important Questions and Answers

Q1. If

$\frac{1}{2} \space \text{is a root of the quadratic equation} \space x^2-mx-\frac{5}{4}=0 \space \text{then the value of m is:}\\$ Options

(a) 2
(b) -2
(c) -3
(d) 3

Ans. (b)-2
Explanation:
Given
$x=\frac{1}{2} \\ \text{as root of the equation}\\ x^2-mx-\frac{5}{4}=0\\ \therefore \begin{pmatrix}\frac{1}{2} \end{pmatrix}^2-m \\ \begin{pmatrix}\frac{1}{2} \end{pmatrix}-\frac{5}{4}=0\\ \Rightarrow \frac{1}{4}-\frac{m}{2}-\frac{5}{4}=0 \\$ m = - 2

Q2. If the roots of the quadratic equation $2kx^2 + (2a + b)x - ab = 0$ are (–2, a), the value of k is

Options

(a) -1
(b) -2
(c) 1
(d) 2

Ans. (a) -1

Explanation:
$2kx + (2a + b)x – ab = 0\\ \therefore \text{a is a root of this equation}\\ 2ka + 2a + ab – ab = 0\\ 2a(k + 1) = 0 \\ \therefore k = –1$

Q3. Solve the equation $4x^2– 5x – 3 = 0 \space$ and give your answer correct to two decimal places.

Explanation:
Given the equation is,
4x– 5x – 3 = 0
Compared with ax+ bx + c = 0,
we have,
a = 4, b = – 5, c = – 3
$\therefore x=\frac{-b\pm\sqrt{-4ac}}{2a}\\ =\frac{-(-5)+\sqrt{(-5)^2-4×4(-3)}}{2×4}\\ =\frac{5\pm\sqrt{25+48}}{8}=\frac{5\pm\sqrt{73}}{8}=\frac{5\pm8.544}{8}\\ =\frac{5\pm8.544}{8} \space or \space \frac{5-8.544}{8}\\ =\frac{13.544}{8}\space or \space \frac{-3.544}{8} \\ = 1.693 \space or \space – 0.443= 1.69 \space or \space – 0.44 \\$ (correct to 2 decimal places) Ans.

Q4. Solve the equation correctly to two significant figures: $x^2 - 3 (x + 3) = 0.$

Explanation:
Given equation is,
$x^2 – 3 (x + 3) = 0 \\ ⇒ x^2 – 3x – 9 = 0 \\ \text{On comparing the equation with} \\ ax^2 + bx + c = 0, \\ \text{we get,} \\ ∴ a = 1, b = – 3, c = – 9 \\ b^2 – 4ac = (– 3)2 – 4 (1) (– 9)\\ = 9 + 36= 45 \\ x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\ = \frac{-(-3)\pm\sqrt{45}}{2×1} \\ = \frac{3\pm3\sqrt{5}}{2} \\ x =\frac{3+3×2.236}{2} \space and \space \frac{3-3×2.236}{2} \\ x = \frac{3+6.708}{2}\space and\space \frac{3+6.708}{2} \\ x = \frac{9.708}{2} \space and\space \frac{3.708}{2} \\$ x = 4·854 and x = – 1·854
∴ x = 4·9 and x = – 1·9 Ans.

Q5. Ram wishes to start a 200 $m^2$ rectangular vegetable garden. Since he has 50 m barbed wire, he fences three sides of the rectangular garden letting the compound wall act as the fourth side of the fence. Find the dimensions of the garden.

Explanation:
Let ABCD be the rectangular garden and CD acts as his house compound wall.
Let Length AB = x m
Breadth BC = y m
According to the question,
xy = 200 ...(i)
x + 2y = 50 ...(ii)
From equation (ii), we get
x = 50 – 2y
Putting the value of x in equation (i), we get
(50 – 2y)y = 200
$⇒ 50y – 2y^2 = 200 \\ ⇒ 2y^2– 50y + 200 = 0 \\ ⇒ y^2 – 25y + 100 = 0 \\ ⇒ y^2– 20y – 5y + 100 = 0 \\$ ⇒ y(y – 20) – 5(y – 20) = 0
⇒ (y – 20)(y – 5) = 0
So, y = 5, 20

When y = 5, then
x = 50 – 2y
x = 50 – 2 × 5 = 40
When y = 20, then
x = 50 – 2y
= 50 – 2 × 20 = 10
So, dimensions of the garden are 10 m and 20 m or 40 m and 5 m. Ans.

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ICSE Class 10 Maths Chapter wise Important Questions

Chapter No.	Chapter Name
Chapter 1	Goods and Service Tax (GST)
Chapter 2	Banking
Chapter 3	Shares and Dividends
Chapter 4	Linear inequations
Chapter 5	Quadratic Equations in one variable
Chapter 6	Ratio and proportion
Chapter 7	Factorization
Chapter 8	Matrices
Chapter 9	Arithmetic Progression
Chapter 10	Geometric Progression
Chapter 11	Coordinate Geometry
Chapter 12	Reflection
Chapter 13	Similarity
Chapter 14	Loci
Chapter 15	Circles
Chapter 16	Constructions
Chapter 17	Mensuration
Chapter 18	Trigonometry
Chapter 19	Statistics
Chapter 20	Probability

Conclusion

Class 10 Mathematics, Quadratic Equations in one variable are like special puzzles we solve to find hidden treasures. These equations look like

ax^2 +bx+c=0 \space

Where a, b, and c are constants, with a ≠ 0 x is the treasure map. So, as you explore quadratic equations in Class 10, think of them as your treasure hunt, your secret code, and your path to becoming a math detective who can uncover hidden gems in the world of mathematics and beyond. For those looking to enhance their skills and develop a more profound understanding of the concepts discussed in this chapter, oswal.io offers an extensive array of practice questions. These quadratic equation class 10 icse board questions have been specifically crafted to support your pursuit of a deeper comprehension of the subject matter.

Frequently Asked Questions

Q1. What is a quadratic equation in one variable?

Ans: A quadratic equation in one variable is a polynomial equation of the form

ax^2 +bx+c=0, \space

where a, b, and c are constants, and x is the variable.

Chapter Wise Important Questions for ICSE Board Class 10 Mathematics
Goods and Service Tax (GST)
Banking
Shares and Dividends
Linear inequations
Quadratic Equations in one variable
Ratio and proportion
Factorization
Matrices
Arithmetic Progression
Geometric Progression
Coordinate Geometry
Reflection
Similarity
Loci
Circles
Constructions
Mensuration
Trigonometry
Statistics
Probability

Class 10 Maths Chapter 5 Quadratic Equations in one variable Important Questions

Introduction

What are Quadratic Equations in one variable?

Class 10 Quadratic Equations in one variable Important Questions and Answers

Q1. If

\frac{1}{2} \space \text{is a root of the quadratic equation} \space x^2-mx-\frac{5}{4}=0 \space \text{then the value of m is:}\\Options

(a) 2(b) -2(c) -3(d) 3

Q2. If the roots of the quadratic equation 2kx^2 + (2a + b)x – ab = 0 are (–2, a), the value of k is

Options

(a) -1(b) -2(c) 1(d) 2

Q3. Solve the equation 4x^2– 5x – 3 = 0 \space and give your answer correct to two decimal places.

Q4. Solve the equation correctly to two significant figures: x^2 – 3 (x + 3) = 0.

Q5. Ram wishes to start a 200 m^2 rectangular vegetable garden. Since he has 50 m barbed wire, he fences three sides of the rectangular garden letting the compound wall act as the fourth side of the fence. Find the dimensions of the garden.