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
a4
b) product of the roots.
b 1
x  9x  5  0
2
#2:
4x  x  3  0
#4:
c  3
a 1
b9
S 
P 
S
9 

1
9 

 9
1
c
a
a
c5
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 
c2
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