•Intro: We already know the standard
form of a quadratic equation is:
y = ax2 + bx + c
•The constants are: a , b, c
•The variables are: y, x
•The ROOTS (or
solutions) of a
polynomial are its
x-intercepts
•Recall: The xintercepts occur
where y = 0.
Roots
•Example: Find the
roots: y = x2 + x - 6
•Solution: Factoring:
y = (x + 3)(x - 2)
•0 = (x + 3)(x - 2)
•The roots are:
•x = -3; x = 2
Roots
•But what about
NASTY trinomials
that don’t factor?
•Abu Ja'far Muhammad
ibn Musa Al-Khwarizmi
•Born: about 780 in
Baghdad (Iraq)
•Died: about 850
•After centuries of
work,
mathematicians
realized that as long
as you know the
coefficients, you can
find the roots of the
quadratic. Even if it
doesn’t factor!
y  ax  bx  c, a  0
2
b  b  4ac
x
2a
2
Solve: y = 5x  8x  3
a  5, b  8, c  3
8  64  60
x
10
2
b  b  4ac
8

4
x
x
2a
10
2
8 2
(8)  (8)  4(5)(3)
x
x
10
2(5)
2
8 2
x
10
8  2 10
x
 1
10
10
82 6 3
x
 
10
10 5
Roots
Plug in your
answers for x.
If you’re right,
you’ll get y = 0.



 




2
 
 
 
 
y  5 35  8 35  3
y  5 9 25  24 5  3
45
24  3
y


2
25
5
y  5(1)  8(1)  3
9
24
15
y



y  583
5
5
5
y0
y0
7

49

32
Solve : y  2x  7x  4 x 
4
a  2, b  7, c  4
7  81
x
2
b  b  4ac
4
x
7  9 2 1
2a
x
x 
4
4
2
2
(7)  (7)  4(2)(4)
16
x
x
 4
2(2)
4
2
Remember: All the terms must be on one
side BEFORE you use the quadratic
formula.
•Example: Solve 3m2 - 8 = 10m
•Solution: 3m2 - 10m - 8 = 0
•a = 3, b = -10, c = -8
2  4  84
•Solve:
= 7 - 2x
x
6
•Solution: 3x2 + 2x - 7 = 0
2  88
x

•a = 3, b = 2, c = -7
6
2
2  4 • 22
b  b  4ac
x
x
6
2a
2
(2)  (2)  4(3)(7) x  2  2 22
x
6
1  22
2(3)
x
3
3x2
• Evariste
Galois
(bottom
We use the
quadratic
formula
picture)
that there is
to solve showed
second degree
no
universal
formula
for
any
equations.higher
Mathematicians
equations
than the
tried for
300 years
to solve
fourth
degree.
When
Galois
was
20, he wrote
in ONE until
higher-degree
equations
NIGHT
much
the basis for a
Niels Abel
(topofpicture)
new
theory
of
solving
proved
that
no
formula
can
be
equations. Sadly, he was
used toinsolve
allthe
fifth-degree
killed
a duel
next day.
equations.Don’t
He was
• MORAL:
do 22!
your
homework late at night.