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Algebra 2B
Chapter 12 Notes
Chapter 12
12.1 - Functions Involving Square Roots
The Domain is the values of x for which the radicand is non-negative
y= 3
x +1
x
0
y
3
1
3
2
x
3
3
4 =3x2 =6
The Range is the values for y
y
12.2 - Operations with Radical Expressions
a. Adding/subtracting Radicals
If the radicals are the same (i.e. the number inside the square root sign) then
combine the numbers outside the radical. Whenever possible, always simply first.
Ex:
2
Ex:
3
+ 3
3
50
+ 3
2
=
5
=
2
3
.
25 + 3
2
= 5
2
= 8
2
+ 3
2
b. Multiplying Radicals
Multiply the numbers inside the radicals and then simply. In some cases you
need to use the distributive property first.
Ex:
2
.
Ex:
3
(3-
18
=
36
)=3
2
= 6
3
-
3
.
2
= 3
3
-
6
c. Simplifying Radicals
Factor the number inside the radical to find a square. Take the square root of the number
and place it in front of the radical. If a radical is in the denominator of a fraction, multiply
the numerator and denominator by the radical.
Ex:
32 =
Ex:
7
16
.2
= 4
7
5
=
5
2
7
5
=
5
5
5
12.3 – Solving Radical Equations
To remove the radical sign square both sides of the equal sign.
Ex:
x
(
= 5
x
)2 = ( 5 )2
x
Ex:
x+2
- 2 = 5
+2
+2
x +2
= 3
(
=
25
x +2
)2 = ( 3 )2
x + 2 = 9
- 2
-2
x
=7
Ex:
x =
x2
3x + 4
= 3x + 4
x2 - 3x – 4 = 0
(x + 1)(x - 4) = 0
x + 1 =0 & x -4=0
x = -1, 4
12.4 – Rational Exponents
Rewriting using rational exponent notation
Ex:
3
= 31/2
5
7
= 71/5
3
11
= 113/2
Rewrite using radical notation
Ex: 87/3 = (
3
8 )7
Evaluating the expression
Ex: 35/3
.
31/3
= 3 5/3
+ 1/3
= 3 6/3
= 32 = 9
12.5 – Completing the Square
This is done by squaring half of the coefficient of the “x” term or
Ex: Complete the square of x2 + 6x
b = 6 so the last term is (6/2)2 = 32 = 9 so
x2 + 6x becomes x2 + 6x + 9 which equals (x + 3)2
b
( 2 )2
Ex: Now solve by completing the square of x2 - 6x – 3 = 0
x2 - 6x – 3 = 0 becomes x2 - 6x = 3 by adding 3 to both sides, now go through the
process of completing the square like above. So c = (-6/2)2 = 32 = 9, and then
x2 - 6x + 9 = 3 + 9 or x2 - 6x + 9 = 12 and then solve by factoring and taking the
square root of both sides.
x2 - 6x + 9 = 12 becomes (x + 3)(x + 3) = 12 or (x + 3)2 = 3 now take the square root
of both sides of the equal sign, so (x + 3)2 = 12 becomes
(x + 3)2 =
x+3
12 which becomes
= + 2
3 and then subtract 3 from both sides
x =-3+ 2
3
12.6 - Pythagorean Theorem and its Converse
If a triangle is a right triangle, then the sum of the square of the lengths of the legs equals the
square of the length of the hypotenuse.
c
a & b are legs and c is the hypotenuse
and a2 + b2 = c2
a
b
Ex: Give a = 3 and b = 4, find c
32 + 42
9 + 16
25
25
c
3
= c2
= c2
= c2
= c2
4
5 = c
Ex: Given a = x, b = x + 2, and c = 10, find a & b.
x
x +2
10
x2 + (x +2)2 = 102
x2 + (x +2)(x +2) = 102
x2 + x2 + 2x +2x + 4 = 100
2x2+ 4x + 4 = 100
x2 + 2x + 2 = 50
x2 + 2x + 2 = 50 – 2
x2 + 2x + 1 = 48 +1
(x + 1) 2 = 49
(x + 1) 2 = 49
x+1= 7
-1=-1
x = 6
So a = 6 and b = 6 +2 or 8
12.7 & 12.8 – The Distance and Mid-point Formulas
The distance formula is
d =
( x2 - x1)2 + ( y2 - y1)2
Ex: Find the distance between the points (1, 4), (-2, 3).
(-2 - 1)2 + (3 – 4)2
d =
( -3)2 + ( -1)2
=
=
9+1 =
The mid-point formula is
=
10
( x1 + x2) , ( y1 + y2)
2
2
Ex: Find the mid-point between the two points (2, 5), 2, -1)
d = (2 + 2) , (-1 + 5)
2
2
= (4), (4)
2
2
12.9 - Logical Reasoning: Proof
= (2, 2)