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 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  
 
       
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FAQs on Let's Recap: Quadratic Equation - Mathematics (Maths) Class 10

1. What is a quadratic equation?
Ans. A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.
2. How do you solve a quadratic equation by factorization?
Ans. To solve a quadratic equation by factorization, we need to factorize the equation and equate each factor to zero. By setting each factor equal to zero, we can find the values of x that satisfy the equation.
3. Can all quadratic equations be solved by factorization?
Ans. No, not all quadratic equations can be solved by factorization. Some quadratic equations may have complex roots or may not be factorizable. In such cases, we need to use other methods like the quadratic formula or completing the square to solve the equation.
4. What is the quadratic formula?
Ans. The quadratic formula is a formula used to solve quadratic equations of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The formula is given by x = (-b ± √(b^2 - 4ac)) / (2a).
5. How do you find the nature of the roots of a quadratic equation?
Ans. To find the nature of the roots of a quadratic equation, we can look at the discriminant (D) of the equation. If D > 0, the equation has two distinct real roots. If D = 0, the equation has two equal real roots. And if D < 0, the equation has two complex roots.
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