1. Unit :- Quadratic Equations
Presented by
Tushar Chandrakant Mhatre (Asst. Teacher) Shri Chhatrapati Shivaji Highschool and Loknete D. B. Patil junior college Jasai

2. Subunits 1. Standard form of quadratic equation
2. Roots of quadratic equation
3. Methods of solving quadratic equations
4. Nature of roots of a quadratic equation (New)
5. Relation between roots and coefficients (New)
6.Form quadratic equations if its roots are given (New)
7. Equations reducible to quadratic form
8. Word problems on quadratic equations

3. 2.Quadratic Equations What is meant by an equation ?
See the following sentences.
1) = 9
true
5) = 9
false
2) 3 x 4 +1= 13
6) = 11
3)10-2 =8
7) = 7
4)12 ÷ 4 = 3
Statement: - Statement is sentence having definite truth value.

4. For these statements we can not assert true or false.
(1) X + 2 = 9
(2) Y - 7 = 5
For these statements we can not assert true or false.
We can say that, statements are true for certain value of variable. This is type of statement is called open statement.
Variable :- It is quantity that can assume any value from set of numbers. And that set of numbers is called domain of variable.

5. For finding the solution of any equation we
Solution of equation To find the solution of an equation means to find value of variable which satisfy the equation ,by which two sides are equal.
For finding the solution of any equation we
use the properties of equality and find the
simplest form of equivalent equations.
Properties of equality
If a = b ,then
i) a + c = b + c ii) a - c = b - c
iii) a x c = b x c iv) a ÷ c = b ÷ c , c ≠ 0

6. Equivalent equations The equations which are having same solutions are called equivalent equations.
e.g i) x + 2 = 5
ii) 2x + 5 = 11
First method of solving equation is enumeration i.e. substituting values of variable .

7. So we use properties of equalities and find equivalent equations.
Consider equation , 5x x =1 = 2 3
To solve this type of equation by enumeration method is become so lengthy.
So we use properties of equalities and find equivalent equations.
15x - 14x
These three equations are equivalent equations 
= 1 6 x 
= 1 6
 X = 6

8. ii) Linear equation in two variable.
Types of equations
i) Linear equation in one variable. Standard form : ax + b = 0 where a , b are real numbers and a ≠ 0, e.g. 3x +6 = 0
ii) Linear equation in two variable.
Standard form : ax + by + c = 0 where a , b, c are real numbers ,a≠o ,b≠o.
iii) Quadratic equation 2
Standard form : ax + bx + c = 0 where a , b, c are real numbers and a ≠ 0 .

9. Methods of solving quadratic equations
Solve X2 + 4X + 3 = 0
i) Enumeration :- By substituting values of variable find the value of X such that
L.H.S = R.H.S
Equivalent equation
ii)
X2 +4X+3=0
linear equation
If a X b =0,then a=o or b=o
(X+3)(X+1)=0
X+3=0 OR X+1=0
 X = -3 OR X = -1
This is a factorization method.

10. Solve: X2 + 8X + 9 = 0 X2 +8X+9=0  X2 +8X+16-7=0  (X+4)2 - ( 7 ) =0
We can factorize the quadratic polynomial by splitting middle term ,But it is difficult job.
So we adopt different method.
X2 +8X+9=0
 X2 +8X+16-7=0
Equivalent equation
 (X+4)2 - ( 7 ) =0 2 2
a – b
=(a+b) (a-b)
 (X+4)2 -( 7 )2 = 0
 (X )(X+4- 7) =0
 (X )=0 or (X ) =0
 X= or X=-4+ 7
This method is called perfect square method