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

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:
1. a, b, and c are constants, with a ≠ 0
2. x is the variable we're solving for
3. $$ax^2$$ represents the quadratic term.
4. bx represents the linear term.
5. c is the constant term.
6. 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.

## 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:
1. a, b, and c are constants, with a ≠ 0
2. x is the variable we're solving for
3. $$ax^2$$ represents the quadratic term.
4. bx represents the linear term.
5. c is the constant term.
6. 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

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

Ans. (b)-2

Explanation:
Given
$$x=\frac{1}{2}$$
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

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

Ans. (a) -1

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

#### Q3. Solve the equation $$4x^2– 5x – 3 = 0$$ 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
$$\therefore x=\frac{-b\pm\sqrt{-4ac}}{2a}\\ =\frac{-(-5)+\sqrt{(-5)^2-4×4(-3)}}{2×4}\\ =\frac{5\pm\sqrt{25+48}}{8}=\frac{5\pm\sqrt{73}}{8}=\frac{5\pm8.544}{8}\\ =\frac{5\pm8.544}{8}or \frac{5-8.544}{8}\\ =\frac{13.544}{8}or\frac{-3.544}{8}$$
= 1.693 or – 0.443
= 1.69 or – 0.44 (correct to 2 decimal places) Ans.

#### Q4. Solve the equation correctly to two significant figures: x2 – 3 (x + 3) = 0.

Explanation:
Given equation is, x2 – 3 (x + 3) = 0
⇒ x2 – 3x – 9 = 0
On comparing the equation with ax2 + bx + c = 0, we get
∴ a = 1, b = – 3, c = – 9
b2 – 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 =$$\frac{3+3×2.236}{2}$$ and $$\frac{3-3×2.236}{2}$$
x =$$\frac{3+6.708}{2}$$and$$\frac{3+6.708}{2}$$
x =$$\frac{9.708}{2}$$and$$\frac{3.708}{2}$$
x = 4·854 and x = – 1·854
∴ x = 4·9 and x = – 1·9 Ans.

#### Q5. Ram wishes to start a 200 m2 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:
et ABCD be the rectangular garden and CD acts as his house compound wall
Let Length AB = x 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 – 2y2 = 200
⇒ 2y2– 50y + 200 = 0
⇒y2 – 25y + 100 = 0
⇒ y2– 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.

#### 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 ax2 +bx+c=0 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.

#### Q1 : What is a quadratic equation in one variable?

Ans: A quadratic equation in one variable is a polynomial equation of the form ax2 +bx+c=0, where a, b, and c are constants, and x is the variable.

#### 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.

#### 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.

#### 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.

#### 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.

## Chapter Wise  Important Questions for ICSE Board Class 10 Mathematics

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