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Square Roots and Solving Quadratics with Square Roots

Square Roots and Solving Quadratics with Square Roots. Review 9.1-9.2. GET YOUR COMMUNICATORS!!!!. Warm Up Simplify. 1. 5 2 2. 8 2. 64. 25. 225. 144. 3. 12 2 4. 15 2. 400. 5. 20 2. Perfect Square. A number that is the square of a whole number

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Square Roots and Solving Quadratics with Square Roots

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  1. Square Roots and Solving Quadratics with Square Roots Review 9.1-9.2

  2. GET YOUR COMMUNICATORS!!!!

  3. Warm Up Simplify. 1. 522. 82 64 25 225 144 3. 1224. 152 400 5. 202

  4. Perfect Square • A number that is the square of a whole number • Can be represented by arranging objects in a square.

  5. Perfect Squares

  6. Perfect Squares • 1 x 1 = 1 • 2 x 2 = 4 • 3 x 3 = 9 • 4 x 4 = 16

  7. Perfect Squares • 1 x 1 = 1 • 2 x 2 = 4 • 3 x 3 = 9 • 4 x 4 = 16 Activity: Calculate the perfect squares up to 152…

  8. Perfect Squares • 9 x 9 = 81 • 10 x 10 = 100 • 11 x 11 = 121 • 12 x 12 = 144 • 13 x 13 = 169 • 14 x 14 = 196 • 15 x 15 = 225 • 1 x 1 = 1 • 2 x 2 = 4 • 3 x 3 = 9 • 4 x 4 = 16 • 5 x 5 = 25 • 6 x 6 = 36 • 7 x 7 = 49 • 8 x 8 = 64

  9. Activity:Identify the following numbers as perfect squares or not. • 16 • 15 • 146 • 300 • 324 • 729

  10. Activity:Identify the following numbers as perfect squares or not. • 16 = 4 x 4 • 15 • 146 • 300 • 324 = 18 x 18 • 729 = 27 x 27

  11. Perfect Squares: Numbers whose square roots are integers or quotients of integers.

  12. Perfect Squares • One property of a perfect square is that it can be represented by a square array. • Each small square in the array shown has a side length of 1cm. • The large square has a side length of 4 cm. 4cm 4cm 16 cm2

  13. Perfect Squares • The large square has an area of 4cm x 4cm = 16 cm2. • The number 4 is called the square root of 16. • We write: 4 = 16 4cm 4cm 16 cm2

  14. Square Root • A number which, when multiplied by itself, results in another number. • Ex: 5 is the square root of 25. 5 = 25

  15. Finding Square Roots • We can think “what” times “what” equals the larger number. 36 = ___ x ___ -6 -6 6 6 Is there another answer? SO ±6 IS THE SQUARE ROOT OF 36

  16. Finding Square Roots • We can think “what” times “what” equals the larger number. 256 = ___ x ___ 16 -16 -16 16 Is there another answer? SO ±16 IS THE SQUARE ROOT OF 256

  17. Estimating Square Roots 25 = ?

  18. Estimating Square Roots 25 = ±5

  19. Estimating Square Roots - 49 = ?

  20. Estimating Square Roots - 49 = -7 IF THERE IS A SIGN OUT FRONT OF THE RADICAL THAT IS THE SIGN WE USE!!

  21. Estimating Square Roots 27 = ?

  22. Estimating Square Roots 27 = ? Since 27 is not a perfect square, we will leave it in a radical because that is an EXACT ANSWER. If you put in your calculator it would give you 5.196, which is a decimal apporximation.

  23. Estimating Square Roots • Not all numbers are perfect squares. • Not every number has an Integer for a square root. • We have to estimate square roots for numbers between perfect squares.

  24. Estimating Square Roots • To calculate the square root of a non-perfect square 1. Place the values of the adjacent perfect squares on a number line. 2. Interpolate between the points to estimate to the nearest tenth.

  25. Estimating Square Roots • Example: 27 What are the perfect squares on each side of 27? 25 30 35 36

  26. Estimating Square Roots • Example: 27 half 5 6 25 30 35 36 27 Estimate 27 = 5.2

  27. Estimating Square Roots • Example: 27 • Estimate: 27 = 5.2 • Check: (5.2) (5.2) = 27.04

  28. 49 = 7 225 = 15 49 = –7 225 = –15 100 = 10 100 = –10 Find the two square roots of each number. A. 49 7 is a square root, since 7 • 7 = 49. –7 is also a square root, since –7 • –7 = 49. B. 100 10 is a square root, since 10 • 10 = 100. –10 is also a square root, since –10 • –10 = 100. C. 225 15 is a square root, since 15 • 15 = 225. –15 is also a square root, since –15 • –15 = 225.

  29. 25 = 5 289 = 17 25 = –5 289 = –17 144 = 12 144 = –12 Find the two square roots of each number. A. 25 5 is a square root, since 5 • 5 = 25. –5 is also a square root, since –5 • –5 = 25. B. 144 12 is a square root, since 12 • 12 = 144. –12 is also a square root, since –12 • –12 = 144. C. 289 17 is a square root, since 17 • 17 = 289. –17 is also a square root, since –17 • –17 = 289.

  30. Evaluate a Radical Expression EXAMPLE SHOWN BELOW

  31. Evaluate a Radical Expression #1

  32. Evaluate a Radical Expression #2

  33. Evaluate a Radical Expression #3

  34. Evaluate a Radical Expression #4

  35. SOLVING EQUATIONS • SOLVING MEANS “ISOLATE” THE VARIABLE • x = ??? y = ???

  36. Solving quadratics SQUARE ROOT BOTH SIDES • Solve each equation. a. x2 = 4 b. x2 = 5 c. x2 = 0 d. x2 = -1

  37. Solve • Solve 3x2 – 48 = 0 +48 +48 3x2 = 48 3 3 x2 = 16

  38. Example 1: • Solve the equation: 1.) x2 – 7 = 9 2.) z2 + 13 = 5 - 13 - 13 +7 + 7 z2 = -8 x2 = 16

  39. Example 2: • Solve 9m2 = 169 9 9 m2 =

  40. Example 3: • Solve 2x2 + 5 = 15 -5 -5 2x2 = 10 2 2 x2 = 5

  41. Example: 1. 2. 5 5 3 3 x2 = 36 x2 = 25

  42. Example: 3. +6 +6 4x2 = 48 4 4 x2 = 12

  43. Examples: 4. 5. -3 -3 +5 +5 -5x2 = -12 44 -5 -5 x2 = 104 x2 = 12/5

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