Section 8.1 Quadratic Equations The Graphical Connection The Principle of Square Roots Completing...
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Transcript of Section 8.1 Quadratic Equations The Graphical Connection The Principle of Square Roots Completing...
8.1 1
Section 8.1 Quadratic Equations The Graphical Connection The Principle of Square Roots Completing the Square Solving Equations by Completing the Square Problem Solving
8.1 2
Review of Quadratic Equations
.0 and
numbers real are and , , where0
writtenbecan that one isequation quadraticA 2
a
cbacbxax
In Chapter 5, we solved these types of equations by factoring the non-zero side.
When the expression is unfactorable, we need an alternate way to find solutions.
24
0)2)(4(
0862
xorx
xx
xx
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?2
?1243
2
2
2
x
x
xx
8.1 3
Quadratic Functions & Graphs In Chapter 2, we examined
f(x) = ax2 + bx + c a ≠ 0
8.1 4
The Principle of Square Roots
Solve by factoring
x2 – 16 = 0
Then by the square root property
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0)4)(4(
0162
xorx
xx
x
416
16
0162
2
x
x
x
8.1 5
Solve Using the Square Root Property
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18
18
0182
2
x
x
x
x
ix
x
x
x
23
49
492
2
1
094
510
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52
522
2 025
x
x
x
x
8.1 6
General Form of the Principle of Square Roots - Let X be any Expression
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9)2( 2
xorx
xx
x
x
22
2222
22
2)2(
2)(
)2()(
2
2
x
xx
x
x
xfwhenxallfind
xxf
kXorkXthenkXIf ,2
8.1 7
What do you notice about the left expression?
It’s a perfectsquare trinomial:
23
23
2)3(
2962
2
x
x
x
xx
8.1 8
Completing the Square- A technique used to find solutions to quadratic equations
Recall the patterns for factoring “perfect square trinomials:
Add a number to x2 + 6x to make it a perfect square. 222
21 )3(969336 xxxsoand
225
4252 )(5 aaa
8.1 9
Using the Perfect Square Technique in an Equation
solutionsThexx
sidesbothofrootSqxandx
formsquareinWritex
sidesbothtovalueAddxx
sideothertotermconstMovexx
xx
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49)11(
1217212122
7222
07222
2
2
2
2
8.1 10
Solving by Completing the Square
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2
2
)(
)()(
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0273
x
x
x
x
xx
xx
xx
xx
8.1 11
Examples - board Solve by factoring
Use square root property
Use completing the square
8.1 12
Application – Compound Interest
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)1(
)1(
)1(40004410
)1(
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2
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2
r
r
r
r
r
r
rPA t
8.1 13
What Next? The Quadratic Formula Present Section 8.2