CBSE Class 10 Maths Revision Notes for Quadratic Equations

 Quadratic Equations

1. Introduction to Quadratic equation
   
   If p(x) is a quadratic polynomial, then p(x) = 0 is called a quadratic equation.
   
   The general or standard form of a quadratic equation, in the variable x, is given by ax² + bx + c = 0, where a, b, c are real numbers and a ≠ 0.
   
   
2. Roots of the quadratic equation
   • The value of x that satisfies an equation is called the zeroes or roots of the equation.
   • A real number α is said to be a solution/root of the quadratic equation ax² + bx + c = 0 if aα² + bα + c = 0.
   • A quadratic equation has at most two roots.
   
   
3. A quadratic equation can be solved by following algebraic methods:
   i. Splitting the middle term (factorization)
   ii. Completing squares
   iii. Quadratic formula
   
   
4. Splitting the middle term (or factorization) method
   • If ax² + bx + c, a ≠ 0, can be reduced to the product of two linear factors, then the roots of the quadratic equation ax² + bx + c = 0 can be found by equating each factor to zero.
   • Steps involved in solving quadratic equation by splitting the middle term (or factorization) method:
   
   Step 1: Find the product ac.
   
   Step 2: Find the factors of ‘ac’ that add to up to b, using the following criteria:
   i. If ac > 0 and b > 0, then both the factors are positive.
   ii. If ac > 0 and b < 0, then both the factors are negative.
   iii. If ac < 0 and b > 0, then larger factor is positive and smaller factor is negative.
   iv. If ac < 0 and b < 0, then larger factor is negative and smaller factor is positive.
   
   Step 3: Split the middle term into two parts using the factors obtained in the above step.
   
   Step 4: Factorize the quadratic equation obtained in the above step by grouping method. Two factors will be obtained.
   
   Step 5: Equate each of the linear factors to zero to get the value of x.
   
   
5. Completing the square method
   
   • Any quadratic equation can be converted to the form (x + a)² – b² = 0 or (x – a)² + b² = 0 by adding and subtracting the constant term. This method of finding the roots of quadratic equation is called the method of completing the square.
   • The steps involved in solving a quadratic equation by completing the square, are as follows:
   
   Step 1: Make the coefficient of x² unity.
   
   Step 2: Express the coefficient of x in the form 2 × x × p.
   
   Step 3: Add and subtract the square of p.
   
   Step 4: Use the square identity (a + b)² or (a – b)² to obtain the quadratic equation in the required form (x + a)² - b² = 0 or (x – a)² + b² = 0. 
   
   Step 5: Take the constant term to the other side of the equation.
   
   Step 6: Take the square root on both the sides of the obtained equation to get the roots of the given quadratic equation.
   
   
6. Quadratic formula
   
   The roots of a quadratic equation ax² + bx + c = 0 (a ≠ 0 )can be calculated by using the quadratic formula:
   
   
   
   If b² – 4ac < 0, then equation does not have real roots.
   
   
7. Discriminant of a quadratic equation
   
   For the quadratic equation ax² + bx + c = 0, a ≠ 0, the expression b² - 4ac is known as discriminant.
   
   
8. Nature of the roots of a quadratic equation:
   i. If b² – 4ac > 0, the quadratic equation has two distinct real roots.
   ii. If b² – 4ac = 0, the quadratic equation has two equal real roots.
   iii. If b² – 4ac < 0, the quadratic equation has no real roots.
   
   
9. There are many equations which are not in the quadratic form but can be reduced to the quadratic form by simplifications.
   
   
10. Application of quadratic equations
    
    • The applications of quadratic equation can be utilized in solving real life problems.
    • Following points can be helpful in solving word problems:
    i. Every two digit number ‘xy’ where x is a ten’s place and y is a unit’s place can be expressed as xy=10x+y .
    
    ii. Downstream: It means that the boat is running in the direction of the stream
    Upstream: It means that the boat is running in the opposite direction of the stream
    Thus, if
    Speed of boat in still water is x km/h
    And the speed of stream is y km/h
    Then the speed of boat downstream will be (x + y) km/h and in upstream it will be (x − y) km/h.
    
    iii. If a person takes x days to finish a work, then his one day's work =