Sum and Product of the roots
Download
Report
Transcript Sum and Product of the roots
Sum and Product of the roots
Steps:
1. Set the equation equal to 0
2. Determine a,b and c
S
b
a
P
c
a
3. Use appropriate formula
Sum of the
roots!
Product of
the roots!
Page 3
For the given quadratic equations, find
a) sum of the roots and
2
a4
b) product of the roots.
b 1
x 9x 5 0
2
#2:
4x x 3 0
#4:
c 3
a 1
b9
S
P
S
9
1
9
9
1
c
a
a
c5
S
b
P
S
a
5
1
P 5
b
S
1
4
S
1
4
P
c
a
P
3
4
Page 3
#10:
2m 2 5m
2
5m
5m
Write a quadratic equation given the
sum and the product. To do this, we
use the equation:
2m 5m 2 0
2
x Sx P 0
2
a 2
b 5
S
c2
b
a
S
5
2
P
c
a
Sum of
roots
Product
of roots
16 , product
#14: sum 16
80
x Sx P 0
2
P
2
x 16 x 80 0
2
2
x 16
16x 80 0
2
S
5
2
P 1
Notice the sign of the sum changes!
The sign of the product stays the same!
sum 6 , product 8
#16:
r1 r2
#18:
x Sx P 0
2
x 6 x 8 0
5
2
, r1 r2 1
x Sx P 0
2
2
5
x x 1 0
2
5
2
x x 1 0
2
2
x 6x 8 0
2
You may not see this answer
written this way on a test, so
lets rewrite it.
2 x
2
5
2 x 2 1 2 0
2
2x 5x 2 0
2
Page 3
Write a quadratic equation whose
roots are given:
Page 3
#20: 2 ,10
sum 2 10 12
product 2 10 20
x Sx P 0
2
x 12 x 20 0
2
x 12 x 20 0
2
Steps:
1. Find the sum of the roots (by
adding them)
2. Find product of the roots (by
multiplying them)
3. Use sum/product equation to
write the equation
Page 3
#22: 8 ,8
sum 8 8 0
product 8 8 64
x Sx P 0
2
x 0 x 64 0
2
x 64 0
2
#24:
3, 3
sum 3
3 0
product 3 3 9 3
x Sx P 0
2
x 0 x 3 0
2
x 3 0
2
2 i,2 i
#30:
Page 3
#32:
2
3 ,2
sum 2 i 2 i 4
sum 2
product 2 i 2 i 5
product 2
3
3 2
3 2
x Sx P 0
x Sx P 0
x 4 x 5 0
x 4 x 1 0
2
2
x 4x 5 0
2
2
2
x 4x 1 0
2
3 4
3 1
For the given equation, one root is
given. Find the other root.
#38:
x 11 x k 0 , r1 5
2
Page 3
#42:
x kx 16 0 , r1 8
2
x Sx P 0
2
Remember the equation:
The sum is
the number
with the x
term, and
remember
that the
sign
changes!
5 r2 11
x Sx P 0
2
x 11 x k 0
2
S 11 , P k
If we know that one
root is 5, and the sum
of the roots is 11,
then:
r2 6
x kx 16 0
2
S k , P 16
SO
8 r2 16
r2 2
Remember, some roots always come
in pairs:
r1 3 2 i
What does root 2 have to be?
The same is true for the
following:
r1 1
3
r2 1
3
It has to be the conjugate of root 1.
r2 3 2 i
Imaginary roots and radical roots always come in pairs!!
Homework
•Page 3
#3,6,9,13,15,17,19,29
,33,39,41