Points to Remember: Quadratic Equations

 Page 1


Quadratic 
Equations 
Page 2


Quadratic 
Equations 
Quadratic Equations
Factorization Method
Completing The Square
Method
An equation of the form ax
2
 + bx + c = 0, where a, b & c are real numbers & a = 0 is known as a Quadratic Equation   
A Polynomial of degree 2 is known as a Quadratic Polynomial    
Quadratic Formula
Method
ax
2
 + bx + c = 0 ax
2
 + bx + c = 0 ax
2
 + bx + c = 0
Multiply a & c
a x c = h
   Find two numbers
whose product is h and sum is b
    These two numbers
can be a and b or any other (p, q)
    Split the middle term
accordingly &  take out the
common factors
     Finally we’ll get a
product of two linear equations &
hence we can get two values of x
i.e. roots of the equation
Divide the entire equation
by a
    Take the constant c / a
on R. H. S.
    Now add
(    x coecient of x)
2
on both the sides (L.H.S. & R.H.S.) 
    So L.H.S. can be written as
    where b is the
    coecient of x
     Now take square root on
both the sides & then simplify the
linear equation to get roots of x
Step 1 :
Step 2 :
Step 3 :
Step 4 :
Step 5 :
Step 1 :
Step 2 :
Step 4 :
Step 3 :
Step 5 :
1
2
   Simply take a, b & c
with their respective sign
   Find
the two roots of the equation
or
Step 1 :
Step 2 :
Step 3 :
-b ±   b
2
 - 4ac
x =
2a
-b +   b
2
 - 4ac
x =
2a
-b -    b
2
 - 4ac
x =
2a
)
2
(x +    
b
2a
QUADRATIC EQUATIONS 10
Page 3


Quadratic 
Equations 
Quadratic Equations
Factorization Method
Completing The Square
Method
An equation of the form ax
2
 + bx + c = 0, where a, b & c are real numbers & a = 0 is known as a Quadratic Equation   
A Polynomial of degree 2 is known as a Quadratic Polynomial    
Quadratic Formula
Method
ax
2
 + bx + c = 0 ax
2
 + bx + c = 0 ax
2
 + bx + c = 0
Multiply a & c
a x c = h
   Find two numbers
whose product is h and sum is b
    These two numbers
can be a and b or any other (p, q)
    Split the middle term
accordingly &  take out the
common factors
     Finally we’ll get a
product of two linear equations &
hence we can get two values of x
i.e. roots of the equation
Divide the entire equation
by a
    Take the constant c / a
on R. H. S.
    Now add
(    x coecient of x)
2
on both the sides (L.H.S. & R.H.S.) 
    So L.H.S. can be written as
    where b is the
    coecient of x
     Now take square root on
both the sides & then simplify the
linear equation to get roots of x
Step 1 :
Step 2 :
Step 3 :
Step 4 :
Step 5 :
Step 1 :
Step 2 :
Step 4 :
Step 3 :
Step 5 :
1
2
   Simply take a, b & c
with their respective sign
   Find
the two roots of the equation
or
Step 1 :
Step 2 :
Step 3 :
-b ±   b
2
 - 4ac
x =
2a
-b +   b
2
 - 4ac
x =
2a
-b -    b
2
 - 4ac
x =
2a
)
2
(x +    
b
2a
QUADRATIC EQUATIONS 10
Nature of Roots: D = b
2
 - 4ac
D iscriminant is known for discriminating real and non-real roots as well as
to dene nature of roots. 
Conditions:
When  D = 0, roots are real & equal
When  D > 0, roots are real & unequal
When  D < 0, roots are non-real & unequal
 
 Introduction of Quadratic equation 
and Factorization method
Scan the QR Codes to watch our free videos
 Quadratic formula is derived from completing the square 
method
 In word questions, try to nd words like product, area,
Pythagoras theorem, etc; where we will multiply two terms
to obtain a quadratic equation.
 In word problem, the solution thus obtained should always be
checked with conditions given in the question. 
PLEASE KEEP IN MIND
 Speed =      , please remember this formula.
A variety of questions on quadratic equations are based on the
concept of speed.  Also do remember conversions like
 1 hour = 60 minutes, 1 minute = 60 seconds.
?
 To determine whether an equation is quadratic or not, expand 
the given equation,  simplify it as much as possible and then check 
the highest degree.
?
?
?
?
 Distance
Time
QUADRATIC EQUATIONS 11