Transcript Quadratic Equations (二次方程式)

Presentation on : Quadratic
Equations
Submitted by : S.K.GIRI
TGT (Maths)
K.V. NO.1 WRS
RAIPUR SHIFT-2
Sub Topic : Nature of
Roots
In general, a quadratic equation may have :
(1) two distinct (unequal) real roots
(2) one double (repeated) real root
(3) no real roots
D=
2
b -
4ac
Since the expression b2 - 4ac can be
used to determine the nature of the
roots of a quadratic equation in the
form ax2 – bx + c = 0, it is called the
discriminant of the quadratic
equation.
Two distinct (unequal) real roots
D = b2 - 4ac > 0
Two distinct roots
The two roots are given by:
If ax  bx  c  0, a  0
2
 b  b  4ac
x
2a
2
One double (repeated) real roots
D = b2 - 4ac = 0
Repeated Roots
No real roots
D = b2 - 4ac < 0
No real Roots
Relations between the Roots
and the Coefficients
What is the sum of the roots of a
quadratic equation?
• We determined that the general form of the
two roots can be written as:
b  b 2  4ac
b  b 2  4ac
x
or x 
2a
2a
• To find the sum, we add these together
2
2
b

b

4ac

b

b
 4ac
b  b  4ac b  b  4ac 

2a
2a
2a
2

2
b  b 2b b



2a
2a
a

Is there a similar relationship for
the product of the roots?
• Yes! We can use the general form of the
roots to find the product.
b  b 2  4ac b  b 2  4ac
(b  b 2  4ac)(b  b 2  4ac)


2a  2a
2a
2a

b  b b  4ac  b b  4ac  ( b  4ac)
4a 2 
2
2
2
2
4ac c
 2 
4a
a

2
b 2  (b 2  4ac)

4a 2
If α and β are the roots of ax2 + bx +c
= 0,
then sum of roots = α + β
b
  
a
and product of roots = αβ
c
 
a
Forming Quadratic Equations
with Given Roots
Forming Quadratic Equations with Given Roots
In S.3,
when α = 2 and β = -3
x = 2 or x = -3
x – 2 = 0 or x + 3 = 0
(x – 2)(x + 3) = 0
x2 + x – 6 = 0
x2 – (sum of the roots)x + (product of roots) = 0
This "discriminates" or tells us what type of solutions we'll have.
ax  bx  c  0
2
 b  b  4ac
x
2a
2
If we have a quadratic equation and are considering solutions
from the real number system, using the quadratic formula, one of
three things can happen.
1. The "stuff" under the square root can be positive and we'd get
two unequal real solutions if b 2  4ac  0
2. The "stuff" under the square root can be zero and we'd get one
solution (called a repeated or double root because it would factor
2
if
b
 4us
acthe
 0same solution).
into two equal factors, each giving
3. The "stuff" under the square root can be negative and we'd get
no real solutions.
if b 2  4ac  0
The "stuff" under the square root is called the discriminant.
The Discriminant
  b 2  4ac
SUMMARY OF SOLVING QUADRATIC EQUATIONS
• Get the equation in standard form:
ax  bx  c  0
2
• If there is no middle term (b = 0) then get the x2 alone and square
root both sides (if you get a negative under the square root there are
no real solutions).
• If there is no constant term (c = 0) then factor out the common x
and use the null factor law to solve (set each factor = 0).
• If a, b and c are non-zero, see if you can factor and use the null
factor law to solve.
• If it doesn't factor or is hard to factor, use the quadratic formula
to solve (if you get a negative under the square root there are no real
solutions).
END OF THE SLIDE SHOW
“THANKS”