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?