The Square Root
Property
Ifx=c, then
x=vc or x--vc (orxtvc)
If (ax +b) =k , then
ax +b=Vk or ax + b=-Vk.
Solve each equation for
x
after
applying the square root property
1. Solve each equation using the square root property. Find both real and complex solutions.
a)x=24
b) 4y +81=0
2
4y-8
3
Y
Y
)(x-7)=18
X-7 t
18
i
d) S(m +3+7=0
X7
X 7 3
mt
m-St
Completing the Square
To solve a quadratic equation by completing the square:
1.
Rewrite the quadratic equation in the form
2. Add
3.
to
Factor the
4. Solve the
1.
Solve the
solutions.
both sides of the
trinomial
+8X
=
constant
equation.
the left-hand side. It will be
resulting equation using the square
root
a
perfect square trinomial.
property.
following equations by completing the square.
a) *+8x+10 =0
Y+3X
on
x+bx
Find both real and complex
(-16
- lo
b) -2x + 4x+1=00
2-
4)b =-lot+)6
Cx+D
X-
6
tV6
CxX-1
+
2
c)3x-2r+2=0
3
CxX-
x
3
d) 6 x + 5 x - 2 = 4
6xSa 6
6
2
41
4 td
x
gi
I5
3
e)
,
x+1X
-|b
oty
2
Cx+2)
-6
xt
+
The Quadratic Formula
Thesolutions of ax +bx +c=0 where a#0
are
given by the formula
-btb-4ac
X=.
2a
The expression b-4ac is called the discriminant.
.If
b-4ac>0, the equation has two real solutions.
If
b-4ac =0, the equation has one real solution.
If
bs-4ac <0, the equation has two complex solutions.
complex conjugates.
The solutions will be
To solve a quadratic equation using the Quadratic Formula:
1. Write the
2.
Identify
equation in the form ax +bx+c=0.
a,
b,
c
and then substitute the values into the formula.
1.
Solve the
solutions.
following equations using the quadratic formula.
Find both real and
compi
a) 3x6x =8
2
3 - -8-o
X6t
36
5-4Y
63
2-C
6 t 3L1966
136
7
3
6 t 224
2C3
3
6
a l > b s >c>6
b) 4z3(z +1)(z-2)
2
2+212 2-32-6
2
Z
3-1CDM
a CD
-3
a4
-3t S
or
2
o
2x+x-420
P l u g i n C a c u l a t o v
-3
4(2(-D
C341D/4
(2
-3 t
q32
-31
n
Tter
Ans /1
-3 t
a-
d)
X-1
4
O-x
X
t
x+S
eD-YOD)
a
CD
b->cas
2. Use the Pythagorean Theorem to determine the value of x and the measurements Or dn
2
sides of the triangle.
+
A
2X 3/3
+CT)
+x
+41= 1K+2x49
1 4 x +Y4 1512y+9
O 2x - a x o
x20
S
Equations Quodratic
the
"
exponent
in Form
datic
in form
if it
can
be
written
in the form
ax"
the second
of the first term is twice the exponent of
Tosolve an equation
that is quadratic in form,
we
make
a
+bx
+C=0.
Note
that
term.
uSubsttution
w
rewrite the equation like so,
au
Once the
+
bu
resulting equation has been solved for
+c
0
=
u . set
x" equal
and olve
to the solutions
ror
1. Solve the following equations using substitution. Find both real and complex solutions
a) 3-5x -2=0
3u-Su -A=o
(3
+1DCY-2-0
3
N
3
-/
2
b) 3x-5x-8=0
3
x
-
S
x
2
g
:
: D - s N T -5-o
Fadse
- Su -8-6
6
4
r
3
4
64
8o
Soluior-
w
(+
P
t
1