example 1 write the first 20 terms of the following sequence: 1, 4, 9, 16, … perfect squares these...
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Example 1
Write the first 20 terms of the following sequence:
1, 4, 9, 16, …
These numbers are called the Perfect Squares.
x 1 2 3 4 5 6 7 8 910
11
12
13
14
15
16
17
18
19
20
16941 25 36 49 64 81
100
121
144
169
196
225
256
289
324
361
400x2
Square Roots
The number r is a square root of x if r2 = x.• This is usually written • Any positive number has two real square
roots, one positive and one negative, √x and -√x√4 = 2 and -2, since 22 = 4 and (-2)2 = 4
• The positive square root is considered the principal square root
x r
Example 2
Use a calculator to evaluate the following:
1. 2. 3. 4.
3 26
3 2
3 / 2
Example 3
Use a calculator to evaluate the following:
1. 2. 3. 4.
3 25
3 21
Properties of Square Roots
Properties of Square Roots (a, b > 0)
Product Property
Quotient Property
ab a b
a a
b b
18 9 2 3 2
2 2 2
25 525
Simplifying Radicals
Objectives:
1. To simplify square roots
Simplifying Square Root
The properties of square roots allow us to simplify radical expressions.
A radical expression is in simplest form when:
1. The radicand has no perfect-square factor other than 1
2. There’s no radical in the denominator
Simplest Radical Form
Like the number 3/6, is not in its simplest form. Also, the process of simplification for both numbers involves factors.
• Method 1: Factoring out a perfect square.75
75
325
325
35
Simplest Radical Form
In the second method, pairs of factors come out of the radical as single factors, but single factors stay within the radical.
• Method 2: Making a factor tree.
75
25
5
35
3
5
Simplest Radical Form
This method works because pairs of factors are really perfect squares. So 5·5 is 52, the square root of which is 5.
• Method 2: Making a factor tree.
75
25
5
35
3
5
Investigation 1
Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.
12 18 24 32 40
48 60 75 83 3300x
Exercise 4a
Simplify the expression.
27 10 15
9
64
11
25
Exercise 4b
Simplify the expression.
98 8 28
15
4
36
49
Example 5
Evaluate, and then classify the product.
1. (√5)(√5) =2. (2 + √5)(2 – √5) =
Conjugates are Magic!
The radical expressions a + √b and a – √b are called conjugates.
• The product of two conjugates is always a rational number
Example 7
Identify the conjugate of each of the following radical expressions:
1. √72. 5 – √113. √13 + 9
Rationalizing the DenominatorWe can use conjugates to get rid of radicals
in the denominator:
The process of multiplying the top and bottom of a radical expression by the conjugate of the denominator is called rationalizing the denominator.
5
1 31 3
1 3
5 1 3
1 3 1 3
5 5 3
2
5 5 3
2
Fancy One
Exercise 9a
Simplify the expression.
6
5
17
12
6
7 5
1
9 7
Exercise 9b
Simplify the expression.
9
8
19
21
2
4 114
8 3
Solving Quadratics
If a quadratic equation has no linear term, you can use square roots to solve it.
By definition, if x2 = c, then x = √c and x = −√c, usually written x = ±√c– You would only solve a quadratic by finding a
square root if it is of the form
ax2 = c– In this lesson, c > 0, but that does not have to
be true.
Solving Quadratics
If a quadratic equation has no linear term, you can use square roots to solve it.
By definition, if x2 = c, then x = √c and x = -√c, usually written x = √c– To solve a quadratic equation using square
roots:
1. Isolate the squared term
2. Take the square root of both sides
Exercise 10a
Solve 2x2 – 15 = 35 for x.
Exercise 10b
Solve for x.
214 11
3x
The Quadratic Formula
Let a, b, and c be real numbers, with a ≠ 0. The solutions to the quadratic equation ax2 + bx + c = 0 are
2 4
2
b b acx
a
Exercise 11a
Solve using the quadratic formula.
x2 – 5x = 7
Exercise 11b
Solve using the quadratic formula.
1. x2 = 6x – 4
2. 4x2 – 10x = 2x – 9
3. 7x – 5x2 – 4 = 2x + 3
The Discriminant
In the quadratic formula, the expression b2 – 4ac is called the discriminant.
Dis
crim
inan
t
Converse of the Pythagorean Theorem
Objectives:
1. To investigate and use the Converse of the Pythagorean Theorem
2. To classify triangles when the Pythagorean formula is not satisfied
Theorem!
Converse of the Pythagorean Theorem
If the square of the length of the longest side of a triangle is equal to the sum of the squares of the lengths of the other two sides, then it is a right triangle.
Example
Which of the following is a right triangle?
Example
Tell whether a triangle with the given side lengths is a right triangle.
1. 5, 6, 7
2. 5, 6,
3. 5, 6, 861
Theorems!
Acute Triangle Theorem
If the square of the length of the longest side of a triangle is less than the sum of the squares of the lengths of the other two sides, then it is an acute triangle.
Theorems!
Obtuse Triangle Theorem
If the square of the length of the longest side of a triangle is greater than the sum of the squares of the lengths of the other two sides, then it is an obtuse triangle.
Example
Can segments with lengths 4.3 feet, 5.2 feet, and 6.1 feet form a triangle? If so, would the triangle be acute, right, or obtuse?
Example 7
The sides of an obtuse triangle have lengths x, x + 3, and 15. What are the possible values of x if 15 is the longest side of the triangle?