Introduction To Quadratic Equations
Introduction To Quadratic Equations :
We studies about different type of polynomials like linear polynomial (
ax
+
b
),
a
≠ 0 quadratic polynomial (
ax
2
+
bx
+
c
) ,
a
≠ 0, cubic polynomial (
ax
3
+
bx
2
+
cx
+
d
) ,
a
≠ 0 etc. When we equate a quadratic polynomial to zero, we get a quadratic equation i.e.
ax
2
+
bx
+
c
= 0
a
≠ 0 and
a
,
b
,
c
are real numbers and
x
is a variable is called quadratic equation.
For example
Problem For Practice (Definition Of Quadratic Equations)
1.
Find which of the following is/are quadratic equation?
(i)
x
+ 1 = 0
(ii)
(iii)
(iv)
(v)
Uses Of Quadratic Equations In Real Life
Uses Of Quadratic Equations In Real Life :
We used quadratic equation in many real life situations for example your school wants to make a Indore badminton court having area 300 square meter with its length one meter more than twice is breadth. What should be length and breath of the hall.
Suppose breath of the hall is
x
meters so its length should be 2
x
+ 1.
So area of court is
This represent a quadratic equation. Breath of court should be satisfy the equation.
History Behind Quadratic Equation
History Behind Quadratic Equation :
Our knowledge Babylonians were first to solve quadratic equations. They knew how to find two positive numbers with a given positive sum and given a positive product this type problem related to quadratic equations. Solving of quadratic equation, in general form is after created goes to ancient Indian mathematicians. Brahmagupta gave a explicit formula to solve quadratic equation.
Later Sridharacharya derived a formula also known as Sridharacharya method which we are going to use in this chapter many times.
Roots Of Quadric Equation
Roots Of Quadric Equation :
Let us consider a quadratic equation
x
2
+3x+2=0
If we put
in L.H.S then we get
Then we say that
is the roots of the equation
x
2
+3x+2=0
In other words we say that
is a solution of the quadratic equation.
Note:
Zeros of quadratic polynomial
and the roots of quadratic equation
are same.
Factorization Method To Solve Quadratic Equation
Factorization Method To Solve Quadratic Equation :
Let consider a quadratic equation
Step 1:
Find two numbers such that their product is equal to
ac
and their sum equal to
b
Step 2:
Split the middle term using these two numbers.
Step 3:
Take the common factor and simplify
Factorization Method To Solve Quadratic Equation Example 1
Factorization Method To Solve Quadratic Equation Example 1 :
Solve the quadratic equation
using factorization method
Sol.
Give quadratic equation
Step 1:
Find two number such that their product is 14 and their sum is 9 so these two numbers are 2 and 7
Step 2:
Now split the middle term in two numbers
Step 3:
Now take common factor
or
or
Hence
x
= –2 and
x
= –7 are roots of given quadratic equation.
Example 2:
Solve the equation by factorisation method
Sol.
Consider the given equation
Now
⇒
⇒
⇒
or
Hence
and
x
= 4 are solutions of given equation.
Problem For Practice (Factorization Method)
1.
Find the solutions of the following quadratic equation using factorization method
(i)
(ii)
(iii)
(iv)
(v)
Quadratic Formula
Quadratic Formula :
For quadratic equation
,
a
,
b
,
c
∈
R
or
where
known as discriminant denote by D
Proof of quadratic formula using completing square method
Let consider the quadratic equation
Divide the equation by coefficient of
x
2
i.e.
a
.
Subtract
c
a
from both side of this equation
Now use complete square method adding both side
⇒
⇒
⇒
⇒
⇒
or
Example 1:
Find the roots of quadratic equation
Sol.
Consider the given quadratic equation
Compare with standard equation
Now
a
= 2,
b
= 4,
c
= 1
Now using quadratic formula
or
or
Hence roots of quadratic equation are
and
Example 2:
Find the roots of quadratic equation
Sol.
Consider the given equation.
Compare with standard equation
We find that
a
= 1,
b
= 4,
c
= –5
Now using quadratic formula
Putting values of
a
,
b
and
c
So,
or
x
= 1 or
x
= –5
x
= 1 and
x
= –5 are roots of equation.
Example 3:
Find the roots of quadratic equation
Sol.
Consider the given equation
Compare with standard equation
find
a
,
b
,
c
a
= 1,
b
= 1,
c
= 1
Using quadratic formula
or
Hence roots of equation are
and
Problem For Practice (Quadratic Formula)
1.
Find the roots of following quadratic equations
(i)
(ii)
(iii)
(iv)
