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```					•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      Roots
polynomial are its
x-intercepts
•Recall: The x-
intercepts occur
where y = 0.
•Example: Find the
roots: y = x2 + x - 6   Roots
•Solution: Factoring:
y = (x + 3)(x - 2)
•0 = (x + 3)(x - 2)
•The roots are:
•x = -3; x = 2
NASTY trinomials
that don’t factor?
ibn Musa Al-Khwarizmi
•After centuries of
work,
mathematicians
realized that as long
as you know the
coefficients, you can
2
find the roots of the   y  ax  bx  c, a  0
doesn’t factor!            b  b  4ac
x
2a
Solve: y = 5x  8x  3
2
8  64  60
a  5, b  8, c  3     x
10
2
b  b  4ac                  8 4
x                            x
2a                        10
2
(8)  (8)  4(5)(3)          8 2
x                             x
2(5)                    10
8 2
x
10
8  2 10
x       1
10   10
82 6 3
x       
10 10 5
Roots
         
2
Plug in your          y  5 35  8 35  3
If you’re right,                 
y  5 9 25  24 5  3
you’ll get y = 0.
2
y  5(1)  8(1)  3
          
y  45 25  24 5  3

y  583                       
y  9 5  24 5  15 5
y0                                   y0
2
Solve : y  2x  7x  4 x  7  49  32
4
a  2, b  7, c  4
7  81
2          x
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
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:  3x2  = 7 - 2x            x
6
•Solution: 3x2 + 2x - 7 = 0           2  88
•a = 3, b = 2, c = -7             x
6
2
b  b  4ac                      2  4 • 22
x                                x
2a                                6
(2)  (2)  4(3)(7) x  2  2 22
2
x                               6      1  22
2(3)                    x
3
We use Galois (bottom
to solve second that there is
picture) showeddegree
no universal formula for any
equations. Mathematicians
equations higher than the
tried for 300 years to Galois
fourth degree. Whensolve
was 20, he wrote in ONE until
higher-degree equations
NIGHT much of the basis for a
Niels Abel (top picture)
new theory of solving can be
no formula
equations.
used to solve all fifth-degree
killed in a duel the next day.
equations. He was your
• MORAL: Don’t do 22!
homework late at night.

```
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