# CBSE Class 10 Maths Revision Notes for Quadratic Equations

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**Quadratic Equations**

**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 ax^{2}+ 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 ax^{2}+ bx + c = 0 if aα^{2}+ 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 ax^{2}+ bx + c, a ≠ 0, can be reduced to the product of two linear factors, then the roots of the quadratic equation ax^{2}+ 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}– b^{2}= 0 or (x – a)^{2}+ b^{2}= 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 x^{2}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}- b^{2}= 0 or (x – a)^{2}+ b^{2}= 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 ax^{2 }+ bx + c = 0 (a ≠ 0 )can be calculated by using the**quadratic formula:**

If b^{2}– 4ac < 0, then equation does not have real roots.**Discriminant of a quadratic equation**For the quadratic equation ax^{2}+ bx + c = 0, a ≠ 0, the expression b^{2}- 4ac is known as**discriminant.****Nature of the roots**of a quadratic equation:

i. If b^{2}– 4ac > 0, the quadratic equation has**two distinct real roots.**ii. If b^{2}– 4ac = 0, the quadratic equation has**two equal real roots.**iii. If b^{2}– 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

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 =

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