Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots. Using your examples above, answer the following:

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1 Integration Unit 5 Quadratic Toolbox 1: Working with Square Roots Name Period Objective 1: Understanding Square roots Defining a SQUARE ROOT: Square roots are like a division problem but both factors must be identical. The answer is given as the single factor. Answer the following division problems to find a square root of 16: Using your examples above, answer the following: 16 = Defining a PERFECT SQUARE: If the number under the radical sign has identical factors, it is called a perfect square. Exercise: Finding Square roots of Perfect Squares Find the square roots of the following perfect squares: a) 100 = b) 36 = c) 0 = d) 9 25 = d) 25 = f) 121 = g) ± 49 = h) 3 2 = i) ( 4 ) 2 = j) 8 2 = k) ( x ) 2 = l) x 2 = Types of real numbers: Rational numbers: Decimals/numbers that STOP or have a repeating pattern. Irrational Numbers: Decimals/numbers that go on forever without a pattern Exercise: Identifying rational and irrational numbers: Use a calculator to take the square root of the following then identify the number as either rational or irrational (circle one) a) 2 = rational or irrational b) 5 = rational or irrational c) 4 25 = rational or irrational d) 9 = rational or irrational 16 4 e) = rational or irrational f) 0.49 = rational or irrational 5 g) 5.2 rational or irrational h) rational or irrational i) 5 rational or irrational j) 2 rational or irrational 5 Page 1

2 Objective 2: Simplifying real Square Roots Perfect Squares: If the number under a square root sign is a perfect square number (e.g. 25), o There is only 1 part. Take the square root and write the answer without the radical sign. This is the simplified answer. Example: 25 = 5 Non- Perfect Squares: If the square root is NOT a perfect square (e.g. 24) simplify the square roots into two parts. First part: largest perfect square factor (rational root) Second part: a non- perfect square factor (irrational root) Simplify the perfect square root. Put that number OUTSIDE the radical sign. Note: it is STILL multiplied so it should be immediately to the left of the radical sign without any symbol after it. Leave the irrational square root under the radical sign. Example: 24 = 4 i 6 = 4 i 6 = 2 6 Optional check: Enter your ORIGINAL square root problem in your calculator and see the decimal. Enter your simplified radical answer into the calculator. The two decimal answers should match! Simplify the following square roots: 1) 125 2) 32 3) ) ) 28 6) 27 7) ) ± ) 3 ± 50 10) 1± 25 11) ) 3 ± 100 Page 2

3 Objective 2 Practice: Simplifying real Square Roots Page 3

4 Objective 3: Imaginary Numbers and Square Roots Definining an Imaginary Number Although 25 = 5, we can t simplify 25 because no two numbers that are the same will multiply to a negative number. Whenever there is a negative under the square root sign, the problem will appear impossible to simplify. Since 1 is the impossible part, we will make the following definition of 1 : Definition: 1 = i where i represents the imaginary answer to 1 By using the letter i we can work with our answers, knowing they don t really exist. We call these types of square roots imaginary numbers. Simplifying square roots with imaginary answers Negative Perfect Squares: If the number under a square root sign is a negative perfect square number (e.g. - 25), factor the ONE square root into two parts: First part: largest perfect square factor (rational root) Second part: 1 Simplify the perfect square root and write the letter i next to it on the right. This will be your simplified answer. Example: 25 = 1i 25 = 1 i 25 = 5i Negative Non- Perfect Squares: If the square root is NOT a perfect square (e.g. - 24) we will need to simplify the square roots into three parts, one of which is 1. First part: largest perfect square factor (rational root) Second part: a non- perfect square factor (irrational root) Third part: 1 Simplify the perfect square root and write the letter i next to it on the right. Leave the irrational square root under the radical sign. Example: 24 = 1i 4 i 6 = 1 i 4 i 6 = 2i ± 15 6) ) ± 100 Page 4

5 Objective 2 & 3 Practice: Simplifying Real and Imaginary Square Roots Page 5

6 Objectives 2 & 3 Practice: Real and Imaginary Square Roots Level 2 Simplify the following square roots COMPLETELY. Use i in your answer if it is imaginary. 1) 250 2) 0 3) 4 4) 6 4 5) 48 6) 48 7) ) 28 9) ) ) ) 3 ± 50 13) ) ) 3 ± 49 16) 5 ± 81 Answers (scrambled): i 7 3 ± 5i i i 3 4 and 14 3 ± 7i 3 ± 5i 2 5i Page 6

7 Objective 4: Solving Equations Using Square roots (TYPE 1) Solving Quadratic Equations: Any equation containing an x 2 is called quadratic. Depending on the difficulty of the equation, different techniques can be used to solve for x (to get x alone). Inverse operations Method: If there is only an x 2 in the equation (no x), then the problem can be solved using normal solving techniques of inverse operations: o Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: Add or subtract (inverse operation) Then multiply or divide (inverse operation) Then square root to remove the squared (inverse operation)- don t forget to ± See ** for details. Simplify all radicals answers as completely as possible. (objectives #1-3) ** WEIRD MATH FACT: The highest exponent on x in a solve problem tells you how many answers there will be. After taking the square root of both sides, you must ± the side without the variable to get your TWO solutions. So make sure you put the ± sign to get BOTH answers! Example 1: Solve x 2 = 4 Why are there TWO answers to this equation? Example 2: Solve 4x 2 = 100 Example 3: Solve 1 5 x2 = 100 Example 4: Solve 5x 2 = 100 Example 5: Solve x 2 = 30 Page 7

8 Objective 5: Solving using Square Roots (Type 2) Inverse operations Method: o Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: Add or subtract then multiply or divide OUTSIDE of the parentheses Square root to remove the squared (inverse operation)- don t forget to ± If there are parentheses, remove the numbers inside of them LAST by adding or subtracting it. Put the added or subtracted number IN FRONT OF the ±! Simplify all radicals answers completely. (See Objectives #1-3) Example 1: Solve (x 10) 2 = 25 Example 2: 4(x 6) 2 = 64 Objective 4 Quick Practice: Solving using square roots (TYPE 1) Solve the following equations, showing all steps. Remember, each problems will have TWO solutions. CHECK YOUR ANSWERS AS YOU GO!! 1) 3n 2 = 75 2) x 2 12 = 20 3) 2x 2 8 = 192 4) 8x 2 = 128 Objective 5 Quick Practice: Solving using square roots (TYPE 2) 1) 4 = 1 4 (x 5)2 2) 64 = 4(x 5) 2 3) 25 = 5(2x 3) 2 4) 2(5x + 3) 2 = 50 Page 8

9 Objective 4 Practice: Solving Using Square Roots Type 1: Solve the equations showing proper solving steps. Make sure you use the ± sign when taking the square root! Use the puzzle on page 11 to check your answers. SIMPLIFY ALL YOUR ANSWERS COMPLETELY (see Objective #2) 1) x 2 = 81 8) b = 86 2) a 2 = 20 9) 2x 2 3 = 15 3) 3n 2 = 45 10) 5w = 58 4) 7x 2 = 84 11) 4x = 20 5) 2v 2 = 180 6) y 2 49 = 0 12) 7y = 4 7) x 2 16 = 8 13) 7y = 4 Page 9

10 Objective 5 Practice: Solving Using Square Roots Type 2: Solve the equations showing proper solving steps. Make sure you use the ± sign when taking the square root! Use the puzzle on page 11 to check your answers. SIMPLIFY ALL YOUR ANSWERS COMPLETELY (see Objective #2) 14) (a + 3) 2 = 25 18) 5(n +1) 2 = 40 15) (t 4) 2 = 7 19) (2x 3) 2 = 81 16) (x 2) 2 = 28 20) (4t +1) 2 = 49 17) 3(x 5) 2 = 12 Page 10

11 Page 11

12 Objective 6: Solving Quadratic Equations using Square Roots (TYPE 3) Solve by getting rid of the numbers on the same side of the = sign as the x. Do the order of operations backwards: o Add or subtract then multiply or divide OUTSIDE of the parentheses o Square root to remove the squared (inverse operation)- don t forget to ± o If there are parentheses, remove the numbers inside of them LAST by adding or subtracting it. Put the added or subtracted number IN FRONT OF the ±! o Simplify all radicals answers completely. (See Objectives #1-3) Example 1: Solve 5(x 2) = 24 Example 2) 2(x 3) 2 7 = 1 Objective 6 Quick Practice: Solving using square roots (TYPE 3) 1) 5 = 2(x 3) ) (x 10) = 12 3) (y +10) 2 49 = 0 4) 0 = 2(x + 5) ) 5 = 2(x 3) ) (5x + 4) = 40 Page 12

13 Practice ALL TYPES: Hidden Message Card 1 s Letter Copy and solve Card 5 s Letter Copy and solve Card 2 s Letter Copy and solve Card 6 s Letter Copy and solve Card 3 s Letter Copy and solve Card 7 s Letter Copy and solve Card 4 s Letter Copy and solve Card 8 s Letter Copy and solve Page 13

14 Card 9 s Letter Copy and solve Card 13 s Letter Copy and solve Card 10 s Letter Copy and solve Card 14 s Letter Copy and solve Card 11 s Letter Copy and solve Card 15 s Letter Copy and solve Card 12 s Letter Copy and solve Card 16 s Letter Hidden Message Page 14

15 Extension Objective 7: Applying Solving techniques to Quadratic Functions REVIEW: Vertex form of a parabola: y = a(x h) 2 + k where (h,k) is the vertex Vertex Form: y = a(x h) 2 + k is called vertex form because the parent function s main point is called the vertex. The vertex is the top or bottom of the parabola. The vertex is (h,k) and can be seen in the shifted equation y = a(x h) 2 + k. The line of symmetry is a vertical line that goes through the vertex of the parabola. The equation of the line of symmetry is x = h Zeros of a Quadratic Function (Roots or x- intercepts of a graph): The zeros of a function are the values of x when y = 0. On the graph, they are the x- values where the graph crosses the x- axis. Zeros are always found by substituting 0 in for y and solving for x. Example 1) y = (x + 6) Example 2) f (x) = 3(x 3) 2 27 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Graph the equations above using the 5 key points AND the x- intercepts, if any. Page 15

16 Problem Set: 1) y = (x 5) 2 3 2) f (x) = 2(x + 4) 2 10 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Exact answers Decimal values if any: Exact answers Decimal values if any: Graph each equation using the 5 key points AND the x- intercepts, if they exist. Page 16-24

17 3) y = (x + 4) 2 6 4) f (x) = 2(x 3) 2 8 a) Vertex: a) Vertex: b) Line of Symmetry: b) Line of Symmetry: c) Solve for roots: c) Solve for roots: Exact answers Decimal values if any: Exact answers Decimal values if any: Graph each equation using the 5 key points AND the x- intercepts, if they exist Page 17

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