Can we determine a
quadratic equation if we
have its roots?
Do Now: Use the quadratic formula to
determine the general form of BOTH
roots to any quadratic equation.
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:
x
b 
b  4 ac
2
or x 
b 
b  4 ac
2
2a
2a
• To find the sum, we add these together

b 
b  4 ac
2

b 
b  4 ac
2
2a

b 
b  4 ac  b 
2
2a
2a

b  b

2a

 2b
2a

b
a
b  4 ac
2
The sum of the roots is
always the same?
• Yes, for any quadratic equation, due to
the nature of the roots, the sum of the
roots is the opposite of b over a.
• If we know one root and the sum of the
roots, we can find the other root.
• Note: If we know that one root is
imaginary, then the other root is the
CONJUGATE!!!
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  4 ac
2

b 
b  4 ac
2
2a
b  4 ac )( b 
2
2
2
4a
2
2

4 ac
4a
2

c
a

2
b  (b  4 ac )
2
2

2
4a

b  4 ac )
2a  2a
2a
b  b b  4 ac  b b  4 ac  ( b  4 ac )
2


(b 
2
Example
• What are the sum and product of the
roots of the equation 3x2-6x+8=0
• Sum b ( 6) 6


a
3
• Product
c

a

8
3
2
3
Why do we care about the
sum and product of roots?
• If we know the sum and product, we can
write the original quadratic equation.
• The sum is made of b and a, and the
product is made of c and a, so we have
everything we need to write the
quadratic equation.
Example
• Find the quadratic equation whose roots
are:
3

(3 
2 )  (3 
(3 
2 )( 3 
2 and 3 
2)  6
6
1) Find sum and
product
2)
93 2  3 2 
 9  2  7
b
2
 b   6, a  1
4
2) Find a, b, and c
a
7
c
 c  7
a
0  x  6x  7
2
3) Write equation
Try on your own
• Find the quadratic equation whose roots
are 5+2i and 5-2i.
Summary/HW
• How can we determine a quadratic
equation if we have the roots of the
equation?
• HW pg 87, 1-10