Let's Recap: Quadratic Equation | Mathematics (Maths) Class 10 PDF Download

 Page 1


 
 
 
(Quadratic equation is an equation in which the highest power of an 
unknown variable is 2) 
 
1. Quadratic Equation:  
Any equation of the form p(x) = 0, where p(x) is a polynomial of 
degree/power 2, is a quadratic equation.  
 
Example: 
  
 
        , 
   
 
      , 
       
 
  , etc. 
 
 
2. Standard Form of Quadratic Equation:  
When we write the terms of p(x) in descending order of their degrees, 
then we get the standard form of the equation i.e. 
 
  
 
       ,  
 
where a, b, c are real numbers, a ? 0. 
 
 
3. Methods to find roots/solutions of a quadratic 
equation: 
 
a. Factorization method 
b. Completing the square method 
c. Discriminant method 
Page 2


 
 
 
(Quadratic equation is an equation in which the highest power of an 
unknown variable is 2) 
 
1. Quadratic Equation:  
Any equation of the form p(x) = 0, where p(x) is a polynomial of 
degree/power 2, is a quadratic equation.  
 
Example: 
  
 
        , 
   
 
      , 
       
 
  , etc. 
 
 
2. Standard Form of Quadratic Equation:  
When we write the terms of p(x) in descending order of their degrees, 
then we get the standard form of the equation i.e. 
 
  
 
       ,  
 
where a, b, c are real numbers, a ? 0. 
 
 
3. Methods to find roots/solutions of a quadratic 
equation: 
 
a. Factorization method 
b. Completing the square method 
c. Discriminant method 
 
 
 
4. Solution of a Quadratic Equation by Factorization:  
a. A real number a is said to be a root of the quadratic equation   
 
 
      , if   
 
        we can say that x = a is a 
solution of the quadratic equation 
 
Note: 
 
 
     
 
 
 
b. If we factorize   
 
     , a ? 0, into a product of two linear 
factors, then the roots of the quadratic equation   
 
        
can be found by equating each factor to zero. 
Example:  
 
The roots of    
 
       are the values of x for which 
 
(3x – 2)(2x + 1) = 0 
 
(3x – 2) = 0           or     (2x + 1) = 0 
 
                               
 
 
                            
  
 
 
 
 
 
 
 
Zeroes of the quadratic 
polynomial ?? ?? 
 ???? ?? 
Roots of the quadratic 
equation ?? ?? 
 ???? ??   
 
 
 
Page 3


 
 
 
(Quadratic equation is an equation in which the highest power of an 
unknown variable is 2) 
 
1. Quadratic Equation:  
Any equation of the form p(x) = 0, where p(x) is a polynomial of 
degree/power 2, is a quadratic equation.  
 
Example: 
  
 
        , 
   
 
      , 
       
 
  , etc. 
 
 
2. Standard Form of Quadratic Equation:  
When we write the terms of p(x) in descending order of their degrees, 
then we get the standard form of the equation i.e. 
 
  
 
       ,  
 
where a, b, c are real numbers, a ? 0. 
 
 
3. Methods to find roots/solutions of a quadratic 
equation: 
 
a. Factorization method 
b. Completing the square method 
c. Discriminant method 
 
 
 
4. Solution of a Quadratic Equation by Factorization:  
a. A real number a is said to be a root of the quadratic equation   
 
 
      , if   
 
        we can say that x = a is a 
solution of the quadratic equation 
 
Note: 
 
 
     
 
 
 
b. If we factorize   
 
     , a ? 0, into a product of two linear 
factors, then the roots of the quadratic equation   
 
        
can be found by equating each factor to zero. 
Example:  
 
The roots of    
 
       are the values of x for which 
 
(3x – 2)(2x + 1) = 0 
 
(3x – 2) = 0           or     (2x + 1) = 0 
 
                               
 
 
                            
  
 
 
 
 
 
 
 
Zeroes of the quadratic 
polynomial ?? ?? 
 ???? ?? 
Roots of the quadratic 
equation ?? ?? 
 ???? ??   
 
 
 
Note:  
For equations with coefficient of ?? 
 other than 1, divide the whole equation by 
the same number on both the sides to get 1 as the coefficient of ?? 
 and then start 
the process of completing the square. 
 
 
 
5. Solution of a Quadratic Equation by Completing The 
Square: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 4


 
 
 
(Quadratic equation is an equation in which the highest power of an 
unknown variable is 2) 
 
1. Quadratic Equation:  
Any equation of the form p(x) = 0, where p(x) is a polynomial of 
degree/power 2, is a quadratic equation.  
 
Example: 
  
 
        , 
   
 
      , 
       
 
  , etc. 
 
 
2. Standard Form of Quadratic Equation:  
When we write the terms of p(x) in descending order of their degrees, 
then we get the standard form of the equation i.e. 
 
  
 
       ,  
 
where a, b, c are real numbers, a ? 0. 
 
 
3. Methods to find roots/solutions of a quadratic 
equation: 
 
a. Factorization method 
b. Completing the square method 
c. Discriminant method 
 
 
 
4. Solution of a Quadratic Equation by Factorization:  
a. A real number a is said to be a root of the quadratic equation   
 
 
      , if   
 
        we can say that x = a is a 
solution of the quadratic equation 
 
Note: 
 
 
     
 
 
 
b. If we factorize   
 
     , a ? 0, into a product of two linear 
factors, then the roots of the quadratic equation   
 
        
can be found by equating each factor to zero. 
Example:  
 
The roots of    
 
       are the values of x for which 
 
(3x – 2)(2x + 1) = 0 
 
(3x – 2) = 0           or     (2x + 1) = 0 
 
                               
 
 
                            
  
 
 
 
 
 
 
 
Zeroes of the quadratic 
polynomial ?? ?? 
 ???? ?? 
Roots of the quadratic 
equation ?? ?? 
 ???? ??   
 
 
 
Note:  
For equations with coefficient of ?? 
 other than 1, divide the whole equation by 
the same number on both the sides to get 1 as the coefficient of ?? 
 and then start 
the process of completing the square. 
 
 
 
5. Solution of a Quadratic Equation by Completing The 
Square: 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
6. Discriminant:  
A discriminant of a quadratic equation determines whether the 
quadratic equation   
 
        has real roots or not. 
          Note: Check point no. 7: Nature of the roots 
 
DISCRIMINANT =  
 
     
 
7. Quadratic Formula:  
The roots of a quadratic equation   
 
        are given by  
 
   v 
 
    
  
 
 
Where, DISCRIMINANT     (Discriminant =  
 
    ) 
 
 
8. Nature of the roots:  
A quadratic equation   
 
        has 
 
a. Two distinct real roots, if  
 
       
 
b. Two equal roots, if  
 
       
 
c. No real roots, if