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Example 1. Write the first 20 terms of the following sequence: 1, 4, 9, 16, …. 16. 100. 121. 144. 169. 196. 225. 256. 289. 324. 361. 400. 25. 36. 49. x 2. 1. 4. 9. 64. 81. These numbers are called the Perfect Squares . Square Roots.

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**Example 1**Write the first 20 terms of the following sequence: 1, 4, 9, 16, … 16 100 121 144 169 196 225 256 289 324 361 400 25 36 49 x2 1 4 9 64 81 These numbers are called the Perfect Squares.**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, √xand -√x √4 = 2 and -2, since 22 = 4 and (-2)2 = 4 • The 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)**Simplifying Radicals**Objectives: • 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: • The radicand has no perfect-square factor other than 1 • 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 52, 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. • (√5)(√5) = • (2 + √5)(2 – √5) =**Conjugates are Magic!**The radical expressions a + √b and a – √bare called conjugates. • The product of two conjugates is always a rational number**Example 7**Identify the conjugate of each of the following radical expressions: • √7 • 5 – √11 • √13 + 9**Rationalizing the Denominator**We 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. Fancy One**Exercise9a**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 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: • Isolate the squared term • Take the square root of both sides**Exercise 10a**Solve 2x2 – 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 ax2 + bx + c = 0are**Exercise 11a**Solve using the quadratic formula. x2 – 5x = 7**Exercise 11b**Solve using the quadratic formula. • x2 = 6x – 4 • 4x2 – 10x = 2x – 9 • 7x – 5x2 – 4 = 2x + 3**The Discriminant**In the quadratic formula, the expression b2 – 4ac is called the discriminant. Discriminant**Converse of the Pythagorean Theorem**Objectives: • To investigate and use the Converse of the Pythagorean Theorem • 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. • 5, 6, 7 • 5, 6, • 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?

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