Gr. 11 Pre-cal
Mrs. Piche
4.1 SOLVING A QUADRATIC EQUATION GRAPHICALLY
Quadratic Equation - an equation in which one of the terms is squared, and no other term is
raised to a higher power.
Generally: ax +bx+c=0, a^O
The solutions to a quadratic equation are called
_. They correspond to the
^-in+'ere^pfS of the graph or the .Z-.-ero^
.of the quadratic function.
Consider the following graph of the function f[x)=x -2x-3 .
The x-intercepts occur at -\ ^ QL-^-G^. -k
The solution to the equation x -2x-3=Q is
K^ -1 a^fljl X.;- ^
Verify by substitution. /)^= -/J
i-&N s.U
^±1^
0
y ^- ?.y-3
( i^- A^-'^ -3
^"-2^-3
1+^-2
3 a^f3) -3
0
^ -fc --3
^-^ g5
0 L.S-^<>
Note: A quadratic equation can have zero, one, or two solutions.
+VJ^? )l--(4
CrrjL K-^"
hO )C~il)
^\0 'r00 .t-S
cy»A 'ro3+
4vK> <rool£
Example 1: Determine the roots of each equation by graphing. Round to the nearest hundredth.
a) x2+6x+3=0 /VA^< ^^- CL^C.^) ^<^.
ry^-^fcX-h? , ^ ^ u .-.-.-.--t
tx = -^.^s"
|.x___^
\^
L;
~2.-%'rOS ',
^^ -^.^
y'a*-^-»-s
^-°^ (-s^^^4('r.^'^^3
^
^
0
/^i-fc?+-_5
^-O.S-S-)^^('-QSTr)^
o.oo^r"
0,oazsr
L.S-2. fcS.
^s^ ^s
-^ ^ L<Se -^^
V^^ '. ( y ^~ I oj(^^^ ^^ -h?
- Xa ^ -1-'°
b) -x2 =2x+l
\j = ~^^-^x~-l
^s
L^
'\!
^yi- I
-^^
"z.-e^o1
- C-0 '
x---^<i^
^
^ v-«'^ .
ar-<) 1-1
-^+»
'-S-| 5
Example 2: The manager at Suzie's Fashion Store has determined that the function R[x)=600-6x
models the expected weekly revenue, R , in dollars, from sweatshirts as the price changes, where x is
the change in price, in dollars. What price increase or decrease will result in no revenue?
Y\o ^e^^v^^ -=?> ^C^ =-0 (©00 -(^}c^n0
Cs-v-o^?^ y= 6oo-G/.^ a^^l Cat<". -z.-e^os
)( -\,0 <OA < ^-(0 ^W3< X Ad^. t^ c^.^9^-'
~7-^yvf* '.
1 r\ a^LCjo .
(^ +^> pn'c^ it^c ID^ A(O <s^\^G. ^^ tT) ;
y=-io
^=-/° tf^ ^ey/«.^u^ ^3 0
Examples: Solve 3m2-m=-2 bygraphing.
!m^-rw + -^ ^ 0
^UL^\ ^ ^ $n^
-^ _^v+
t^Q SoloL^'<^
Example 4: Two numbers have a sum of 12 and a product of 32. Determine the two numbers.
L ^- K ^ep h-^^ X -^ -^Os
^ ' y. '4.^x. -3&
3^ ^ i's ^->
P^(iu..d( ; ?< C^->) =-3^
l'3-y - x^ -3^ ^ 0
'Z^VQ^'.
)f" g-
~ ^cv ^-^x -3^. ~o
Assignment: 4.1 p.215 #2b, 3a,e, 5,10
|S+^ ^|(tJ olnd ^ 'S [>^ "(g_
oi t^ ^ »". ^. ^n^ tfe ^ 1^^ ^ L/