Class-10 : Maths - Chapter : 4 Quadratic Equations (Ex-4.1 Ex-4.2 Ex-4.3 Ex-4.4)

QUADRATIC EQUATIONS 

In this chapter, students will get to know the standard form of writing a quadratic equation. The chapter goes on to explain the method of solving the quadratic equation through the factorization method and completing the square method. The chapter ends with the topic on finding the nature of roots.

To download Ch-4 (Quadratic Equations) solutions in PDF please click on the link below :

Ex - 4.1 : Download PDF 

Ex - 4.2 : Download PDF 

Ex - 4.3 : Download PDF

Ex - 4.4 : Download PDF 

Note : All the copyright of this PDF content belongs to NBSEguideonline. It is available to all NBSEguideonline users without any subscription. 

Points To Remember 

QUADRATIC EQUATIONS 

The polynomial of degree two is called quadratic polynomial and equation corresponding to a 

quadratic polynomial P(x) is called a quadratic equation in variable x.

Thus, P(x) = ax² + bx + c = 0, a ≠ 0, a, b, c ∈ R is known as the standard form of quadratic 

equation.

There are two types of quadratic equation.

(i) Complete quadratic equation : The equation ax² + bx + c = 0 where a ≠ 0, b ≠ 0,c ≠ 0

(ii) Pure quadratic equation : An equation in the form of ax² = 0, a ≠ 0, b = 0, c = 0

ZERO OF A QUADRATIC POLYNOMIAL

The value of x for which the polynomial becomes zero is called zero of a polynomial

For instance,

1 is zero of the polynomial x² - 2x + 1 because it become zero at x = 1.

SOLUTION OF A QUADRATIC EQUATION BY FACTORISATION

A real number x is called a root of the quadratic equation ax² + bx + c = 0, a 0 if aα² + bα + c = 0. In this case, we say x = α is a solution of the quadratic equation.

NOTE:

1. The zeroes of the quadratic polynomial ax² + bx + c and the roots of the quadratic equation ax² + bx + c = 0 are the same.

2. Roots of quadratic equation ax² + bx + c = 0 can be found by factorizing it into two linear factors and equating each factor to zero.

SOLUTION OF A QUADRATIC EQUATION BY COMPLETING THE SQUARE

By adding and subtracting a suitable constant, we club the x² and x terms in the quadratic 

equation so that they become complete square, and solve for x.

In fact, we can convert any quadratic equation to the form (x + a)² - b² = 0 and then we can 

easily find its roots.

DISCRIMINANT

The expression (b² - 4ac) is called the discriminant of the quadratic equation.

SOLUTION OF A QUADRATIC EQUATION BY DISCRIMINANT METHOD

Let quadratic equation is ax² + bx + c = 0 

Step 1. Find D = (b² - 4ac)

Step 2.

(i) If D > 0, roots are given by x = (-b + √D) / 2a , (-b – √D) / 2a

(ii) If D = 0 equation has equal roots and root is given by x = -b / 2a.

(iii) If D < 0, equation has no real roots.

ROOTS OF THE QUADRATIC EQUATION

Let the quadratic equation be ax² + bx + c = 0 (a ≠ 0).

Thus, if (b² - 4ac) ≥ 0, then the roots of the quadratic

—b ± √(b² — 4ac)/ 2a equation are given by QUADRATIC FORMULA

—b ± √(b² — 4ac) / 2a is known as the quadratic formula which is useful for finding the roots of a quadratic equation.

NATURE OF ROOTS

(i) If b² — 4ac > 0, then the roots are real and distinct.
(ii) If b² — 4ac = 0, the roots are real and equal or coincident.
(iii) If b² — 4ac <0, the roots are not real (imaginary roots)

FORMATION OF QUADRATIC EQUATION WHEN TWO ROOTS ARE GIVEN

If α and β are two roots of equation then the required quadratic equation can be formed as x² — (α + β)x + αβ =0

NOTE : 

Let α and β be two roots of the quadratic equation (ax² + bx + c = 0) then

Sum of Roots : = – the coefficient of x / the coefficient t of x²
⇒ α + β = – b / a
Product of Roots : = constant term / the coefficient of x²
⇒ αβ = c / a

METHOD OF SOLVING WORD PROBLEMS

Step 1: Translating the word problem in to Mathematics form (symbolic form) according to the given condition

Step 2 : Form the word problem into Quadratic equations and solve them.

*****

NBSE Solutions for Class 10 Maths for all Chapters by NBSE Guide Online are provided here. Just click on the chapter wise links given below :

• Chapter 1 Real Numbers

• Chapter 2 Polynomials

• Chapter 3 Pair of Linear Equations in Two Variables

• Chapter 4 Quadratic Equations

• Chapter 5 Arithmetic Progressions

• Chapter 6 Triangles

• Chapter 7 Coordinate Geometry

• Chapter 8 Introduction to Trigonometry

• Chapter 9 Some Applications of Trigonometry

• Chapter 10 Circles

• Chapter 11 Constructions

• Chapter 12 Areas Related to Circles

• Chapter 13 Surface Areas and Volumes

• Chapter 14 Statistics

• Chapter 15 Probability

The solution list comprises all the chapter-wise answers to the questions present in the NBSE book for Class 10 in a very precise and lucid manner, maintaining the objective of textbooks.

*****

Related Links

NBSE Class 10 Book-Keeping & Accountancy notes

NBSE Class 9 Book-keeping & Accountancy notes

NBSE Class 9 Mathematics Solutions