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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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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

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

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

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

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