bellringer part two simplify (m – 4) 2. (5n + 3) 2
TRANSCRIPT
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Bellringer part two
• Simplify• (m – 4)2.
• (5n + 3)2.
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Determine the pattern1
4
9
16
25
36
…
= 12
= 22
= 32
= 42
= 52
= 62
These are perfect squares!
You should be able to list at least the first 15 perfect squares in 30 seconds…
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GO!!!• Perfect squares1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225
How far did you get?
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Perfect Square TrinomialAx2 + Bx + C
• Clue 1: A & C are positive, perfect squares.
• Clue 2: B is the square root of A times the square root of C, doubled.
If these two things are true, the trinomial is a Perfect Square Trinomial and can be
factored as (x + y)2 or (x – y)2.
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General Form of Perfect Square Trinomials
• x2 + 2xy + y2 = (x + y)2
or• x2 – 2xy + y2 = (x - y)2
• Note: When factoring, the sign in the binomial is the same as the sign
of B in the trinomial.
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Just watch and think.• Ex) x2 + 12x + 36• What’s the square root of
A? of C?• Multiply these and double.
Does it = B?• Then it’s a Perfect Square
Trinomial!
• Solution: (x + 6)2
• Ex) 16a2 – 56a + 49• Square root of A? of
C?• Multiply and double…• = B?
• Solution: (4a – 7) 2
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Ex. 1: Determine whether each trinomial is a perfect square trinomial.
If so, factor it.1. y² + 8y + 162. 9y² - 30y + 10
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Example 2: Factoring perfect square trinomials.
• 1) x2 + 8x + 16 2) 9n2 + 48n + 64
• 3) 4z2 – 36z + 81 4) 9g² +12g - 4
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4) 25x² - 30x + 9
5) x² + 6x - 9
6) 49y² + 42y + 36
7) 9m³ + 66m² - 48m
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Review: Multiply (x – 2)(x + 2)
First terms:
Outer terms:
Inner terms:
Last terms:
Combine like terms.
x2 – 4
x -2
x
+2
x2
+2x
-2x
-4
This is called the difference of squares.
x2
+2x-2x-4
Notice the middle terms
eliminate each other!
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Difference of Squares
a2 - b2 = (a - b)(a + b)or
a2 - b2 = (a + b)(a - b)
The order does not matter!!
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4 Steps for factoringDifference of Squares
1. Are there only 2 terms?2. Is the first term a perfect square?3. Is the last term a perfect square?4. Is there subtraction (difference) in the
problem?If all of these are true, you can factor
using this method!!!
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1. Factor x2 - 25When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
No
Yes x2 – 25
Yes
Yes
Yes
( )( )5 xx + 5-
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2. Factor 16x2 - 9When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!
No
Yes 16x2 – 9
Yes
Yes
Yes
(4x )(4x )3+ 3-
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When factoring, use your factoring table.
Do you have a GCF?
Are the Difference of Squares steps true?Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer!(9a )(9a )7b+ 7b-
3. Factor 81a2 – 49b2
No
Yes 81a2 – 49b2
Yes
Yes
Yes
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Factor x2 – y2
1. (x + y)(x + y)
2. (x – y)(x + y)
3. (x + y)(x – y)
4. (x – y)(x – y)
Remember, the order doesn’t matter!
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When factoring, use your factoring table.
Do you have a GCF?
3(25x2 – 4)
Are the Difference of Squares steps true?Two terms?
1st term a perfect square?
2nd term a perfect square?
Subtraction?
Write your answer! 3(5x )(5x )2+ 2-
4. Factor 75x2 – 12
Yes! GCF = 3
Yes 3(25x2 – 4)
Yes
Yes
Yes
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Factor 18c2 + 8d2
1. prime
2. 2(9c2 + 4d2)
3. 2(3c – 2d)(3c + 2d)
4. 2(3c + 2d)(3c + 2d)
You cannot factor using difference of squares because there is no
subtraction!
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Factor -64 + 4m2
Rewrite the problem as 4m2 – 64 so the
subtraction is in the middle!
1. prime
2. (2m – 8)(2m + 8)
3. 4(-16 + m2)
4. 4(m – 4)(m + 4)
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Ex. 3: Factor completely.
2x² + 18
c² - 5c + 6
5a³ - 80a
8x² - 18x - 35
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Ex. 3: Solve each equation.
3x² + 24x + 48 = 049a² + 16 = 56a
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z² + 2x + 1= 16 (y – 8)² = 7
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