Example 1 Write the first 20 terms of the following sequence: 1, 4, 9, 16, … Perfect Squares These numbers are called the Perfect Squares. x x2x2

Square Roots square root The number r is a square root of x if r 2 = 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 2 2 = 4 and (-2) 2 = 4 principal square rootThe positive square root is considered the principal square root

Example 2 Use a calculator to evaluate the following:

Example 3 Use a calculator to evaluate the following:

Properties of Square Roots Properties of Square Roots ( a, b > 0) Product Property Quotient Property

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.

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.

Simplest Radical Form This method works because pairs of factors are really perfect squares. So 5·5 is 5 2, the square root of which is 5. Method 2: Making a factor tree.

Investigation 1 Express each square root in its simplest form by factoring out a perfect square or by using a factor tree.

Exercise 4a Simplify the expression.

Exercise 4b Simplify the expression.

Example 5 Evaluate, and then classify the product. 1.( √5)(√5) = 2.(2 + √5)(2 – √5) =

Conjugates are Magic! conjugates 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.√7 2.5 – √11 3.√13 + 9

Rationalizing the Denominator We can use conjugates to get rid of radicals in the denominator: rationalizing 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. Fancy One 

Exercise 9a Simplify the expression.

Exercise 9b Simplify the expression.

Solving Quadratics If a quadratic equation has no linear term, you can use square roots to solve it. By definition, if x 2 = 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 ax 2 = 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 x 2 = 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 2 x 2 – 15 = 35 for x.

Exercise 10b Solve for x.

The Quadratic Formula Let a, b, and c be real numbers, with a ≠ 0. The solutions to the quadratic equation ax 2 + bx + c = 0 are

Exercise 11a Solve using the quadratic formula. x 2 – 5 x = 7

Exercise 11b Solve using the quadratic formula. 1. x 2 = 6 x – x 2 – 10 x = 2 x – x – 5 x 2 – 4 = 2 x + 3

The Discriminant discriminant In the quadratic formula, the expression b 2 – 4 ac is called the discriminant. Discriminant

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, 8

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?