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1copyright (c) Lynda Greene 2002

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Complete the Square Copyright©2002 Lynda Greene 2copyright (c) Lynda Greene 2002

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Completing the Square CONCEPT: What are we doing and Why? 1. If normal factoring methods don’t work, we can alter the problem to create a new trinomial that can be factored. OR 2. When we need to draw a graph, the equation can be changed into the (factored) Standard Form: Note: Completing the Square is used mainly for graphing parabolas, but it can also be used to graph conic sections, such as circles,ellipses, and hyperbolas which also have 2 nd powers. This process is used on Quadratic (or 2 nd power) trinomials. 3copyright (c) Lynda Greene 2002

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Note: There are two major methods for doing the process of Completing the Square. Completing the Square Definitions: EQUATION: problem with an equal sign (=) in it EXPRESSION: problem that does not have an equal sign in it 2. Different sides - which is in most textbooks ----works only on Equations and then you have to switch back to the “Same Side” method if the problem is an Expression. 1.Same Side - which is what we are doing in this lecture -----works on Equations and Expressions. 4copyright (c) Lynda Greene 2002

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Factor x 2 + 6x – 9 We begin by separating the three terms into two groups. x 2 + 6x The first group has the first two terms in it and will become a new trinomial Completing the Square – 9 The second group will have the constant in it. STEP 1: SEPARATE TERMS 5copyright (c) Lynda Greene 2002

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Factor x 2 + 6x – 9 (x 2 + 6x ) Completing the Square Create another blank space for the “balancing term” STEP 2: CREATE BLANK SPACES FOR THE NEW 3 RD TERM AND THE BALANCING TERM - ___ – 9 + ___ Create a blank space for the new 3 rd term. (Draw parentheses around this new trinomial) Notice that the signs are opposites. 3 RD TERM: + BALANCING TERM: - 6copyright (c) Lynda Greene 2002

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Completing the Square STEP 3: CREATE THE FACTOR (x 2 + 6x ) Write this number in the FACTOR - ___ – 9 + ___ Side Calculation: Take half (x 1/2) of the coefficient of the middle term Write this underneath the trinomial. (This is the FACTOR) Think of it as: Half of positive 6 is positive 3 7copyright (c) Lynda Greene 2002

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Completing the Square STEP 4A: CREATE THE 3 rd TERM (x 2 + 6x ) Side Calculation: Use FOIL to find the new 3rd term - ___ – 9 See lesson on Perfect Square Trinomials and FOIL multiplication for details about this process Write the new 3 rd term in the trinomial 8copyright (c) Lynda Greene 2002

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Completing the Square STEP 4B: WRITE THE BALANCING TERM - ___ – 9 Write the same number in the balancing term 9copyright (c) Lynda Greene 2002

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Completing the Square STEP 5: COMBINE THE CONSTANTS Add (or subtract) the CONSTANT and the BALANCING TERM You can drop either of these two terms if they are zeros. 10copyright (c) Lynda Greene 2002

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Here are some step-by-step examples of typical problems. Factor: x 2 + 8x – 7 Separate into two groups x 2 + 8x – 7 Create spaces for the 3rd term and the balancing term ( + ____) - (____) Create the FACTOR 8 ( ½) = 4 (x + 4) 2 FOIL the factor: It is also the balancing term This is the 3 rd term 11copyright (c) Lynda Greene 2002

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Continuing… Combine the Constants (x 2 + 8x + 16) – 7 - 16 - 23 (x + 4) 2 If you need to draw a graph this is where you stop If you have an equation (=) and need to solve for x, continue working until x is by itself (isolated) 12copyright (c) Lynda Greene 2002

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Factor: x 2 + 10x – 7 (x 2 + 10x + ) – 7 - (x + 5) 2 FOIL: (x + 5)(x + 5) = (x 2 + 10x + 25) 25 (+ 25) 1. Separate terms 2. Create Blanks 3. Create Factor 4A. Foil to find 3 rd term 4B. Fill in blanks 5. Add Constants Fill in the 2 blanks (x + 5) 2 – 7 – 25 (x + 5) 2 – 32 Note: You can take a shortcut and avoid the FOIL step by squaring “half the middle term” (i.e. 13copyright (c) Lynda Greene 2002

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Factor: x 2 - 3x + 2 (x 2 - 3x + ) + 2 - Square it: 9/4(+ 9/4) 1. Separate terms 2. Create Blanks 3. Create Factor 4A. Foil to find 3 rd term, or square it 4B. Fill in blanks 5. Add Constants Fill in the 2 blanks Use “common denominators” to add fractions 14copyright (c) Lynda Greene 2002 “3” won’t divide by 2 evenly, so term is 3 over 2

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Factor: x 2 - x - 4 Square it: 1. Separate terms 2. Create Blanks 3. Create Factor 4A. Foil to find 3 rd term, or square it 4B. Fill in blanks 5. Add Constants Fill in the 2 blanks Use “common denominators” to add fractions 15copyright (c) Lynda Greene 2002

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Note: If there is a coefficient in front of the first term Extra Step*: You must factor it out of the first two terms before completing the square. Factor: 2x 2 + 6x – 7 = 0 2(x 2 + 3x+___) – 7- ___ = 0 Separate terms and draw spaces Take half of the middle coefficient (2x 2 + 6x + ____) – 7 - ____= 0 *Factor the 2 out of the 1 st group 16copyright (c) Lynda Greene 2002

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Factor: 2x 2 + 6x – 7 = 0 2(x 2 + 3x __ ) – 7 - ___ = 0 Square it: So, the third term we created was a 9/4 Write it in both blank spots. + 9/4( 9/4) 17copyright (c) Lynda Greene 2002

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Factor: 2x 2 + 6x – 7 = 0 2(x 2 + 3x ) – 7 = 0 But, the number in the parentheses, 9/4, is being multiplied by 2 So the number we actually added to the left side is not 9/4, It is + 9/4 Now add the constants - 2(9/4) 18copyright (c) Lynda Greene 2002

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Completing the Square: Solving for x 19 copyright (c) Lynda Greene 2002 Simplify the roots and subtract 3//2 from both sides: Take the square root of both sides: Divide by 2: Add 23/2 to both sides: This is the answer, but some books write it as one fraction. Either answer is correct.

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Note: If there is a negative coefficient in front of the first term, you must factor it out of the first two terms and change the signs. Factor: -3x 2 + 6x + 4 = 0 -3(x 2 - 2x + ___) + 4 ( )= 0 Separate and draw spaces (-3x 2 + 6x + ____) + 4 - _____ = 0 Factor the -3 out of the 1 st group This changes the signs on 1 st two terms inside the ( ) SIGNS!!! Since the parentheses has a –(-3) it becomes a +3 -(-3) 20copyright (c) Lynda Greene 2002

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Factor: -3x 2 + 6x + 4 = 0 -3(x 2 - 2x ) + 4 + 3 ( ) = 0+ 1 Square it So, the third term we created was a 1 Write it in the inside both blank spots. + 1 -3(x - 1) 2 Take half of the middle coefficient -3(x - 1) 2 + 4 + 3 = 0 Add the constants -3(x - 1) 2 + 7 = 0 21copyright (c) Lynda Greene 2002

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Completing the Square: Solving for x -3(x - 1) 2 + 7 = 0 Subtract the 7 from both sides: -3 (x - 1) 2 = -7 Take the square root of both sides: Add 1 to both sides: -3 -3 Divide by -3: 22copyright (c) Lynda Greene 2002

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You can’t have a square root on the bottom of a fraction So we have to rationalize the denominator Now our solution is: 23copyright (c) Lynda Greene 2002

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Now depending on your teacher’s preference and the textbook you are using, this may be the way the answer is written. Notice that this answer has a “1” separated from the fraction by the plus-minus sign. This can be changed into a single fraction by changing the “1” into a fraction with a common denominator and then adding the two fractions together. Rewrite the “1” as a fraction Multiply the top and bottom by 3 to get the same denominator as the other fraction Rewrite it as a single fraction orAnswer: 24copyright (c) Lynda Greene 2002

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