CBSE Class 10 Maths Revision Notes for Quadratic Equations
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- 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.
- 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.
- A quadratic equation can be solved by following algebraic methods:
i. Splitting the middle term (factorization)
ii. Completing squares
iii. Quadratic formula
- 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.
- 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.
- 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.
- Discriminant of a quadratic equation
For the quadratic equation ax2 + bx + c = 0, a ≠ 0, the expression b2 - 4ac is known as discriminant.
- 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.
- There are many equations which are not in the quadratic form but can be reduced to the quadratic form by simplifications.
- 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
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 =
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