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CBSE Class 10 Maths Formulas Chapter-04 | Quadratic Equations


Quadratic equation Chapter 4 Class 10

Basic concepts on Quadratic Equation class 10, chapter 4,  Nature of roots, Discriminant, Quadratic Formula, method of completing the square. Complete explanation of quadratic equations
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QUADRATIC EQUATION :- 

An equation whose degree is 2 is called a quadratic equation.
General Quadratic Equation is  ax2 + bx + c = 0 
Here "a" is the coefficient of x2 , 
"b" is the coefficient of x and 
"c" is the constant term. 

Difference between the quadratic equations and quadratic polynomials.

Quadratic equations are very similar to the quadratic polynomials. But they are different from each other because of the following reasons.

Quadratic Equations

Quadratic Polynomials

General Quadratic Equations is  
 ax2 + bx + c = 0

General Quadratic Polynomial is 
P(x) = ax2 + bx + c

Solutions of quadratic equations are called its roots.

Solutions of quadratic polynomials are called its zeroes.



Methods of Solving Quadratic Equations
There are mainly three methods of solving Quadratic equations
1) Factor Method
2) By Quadratic Formula
3) My the method of Completing the Square.

ROOTS:-
Solutions of the quadratic equations are called its roots. A quadratic equation have two roots.

 RELATION BETWEEN ROOTS AND COEFFICIENTS:-

equation

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Quadratic Equation From The Roots:-

x2 – (Sum of roots)x + Product of roots = 0

x2 – Sx + P = 0




Nature Of The Roots Of The Quadratic Equation:-
General quadratic equation is 
 ax2 + bx + c = 0

Discriminant of quadratic equation is denoted by D and is given by
D = b2 - 4ac

If D > 0  then  roots are real and unequal or distinct or different)
If D = 0  then  roots are  real and equal
If D < 0  then  roots are  not real
If D ≥ 0  then  roots are real



Quadratic Formula For Solving The Quadratic Equations:-

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Method of completing the square:-

Write the given quadratic equation
ax2 + bx + c = 0
Make the coefficient of x2 unity

equation
Bring the constant term to the right hand side.

equation

equation

equation

Completing the square 

equation

Taking the square root on both side

equation

Find the value of x

equation

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equation

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Example
Find the roots of  5x2 - 6x - 2 = 0 by the method of completing the square
Given equation is   5x2 - 6x - 2 = 0
D = b2 - 4ac 
⇒ D = (-6)2  - 4 x 5 x -2 = 36+40 = 76 > 0
⇒ Roots are real and distinct
⇒ Roots of Q. E. are exists
Now we apply method of completing the square as follows

Step (1) Make the coefficient of x2 unity ( divide the equation by 5)

equation

Step (2) Bring the constant term to the RHS

equation

Step (3)

equation
 
equation

equation 

Step (4)  Completing the square and find the square root

equation

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equation

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Method of solving the Q. E. by using quadratic formula
Find the roots of 5x2 - 6x - 2 = 0 by the Quadratic Formula
Given equation is 5x2 - 6x - 2 = 0
D = b2 - 4ac
⇒ D = (-6)2  - 4 x 5 x -2 = 36 + 40 = 76 > 0
⇒ Roots are real and distinct
⇒ Roots of Q. E. are exists
Now we apply the quadratic formula












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