Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B...
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Transcript of Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B...
![Page 1: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/1.jpg)
Section 5.1
Quadratic Equations
![Page 2: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/2.jpg)
OBJECTIVES
A Find the greatest common factor (GCF) of numbers.
B Find the GCF of terms.
![Page 3: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/3.jpg)
OBJECTIVES
C Factor out the GCF.
D Factor a four-term expression by grouping.
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DEFINITIONGreatest Common Factor (GCF)
The largest common factor of the integers in a list.
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PROCEDUREFinding the Product
4(x + y) = 4x + 4y
5(a – 2b) = 5a – 10b
2x(x + 3) = 2x2 + 6x
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PROCEDUREFinding the Factors
4x + 4y = 4(x + y)
5a – 10b = 5(a – 2b)
2x2 + 6x = 2x(x + 3)
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DEFINITIONGCF of a Polynomial
The term axn is the GCF of a polynomial if
1. a is the greatest integer that divides each coefficient.
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DEFINITIONGCF of a Polynomial
The term axn is the GCF of a polynomial if
2. n is the smallest exponent of x in all the terms.
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Section 5.1Exercise #2
Chapter 5Factoring
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18 30 2
9 15 3
3 5
GCF = 2 • 3 = 6
Find the GCF of 18x2y4 and 30x3y5 .
x2y4
x3y5
x2y4
GCF = 6x2y4
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Section 5.1Exercise #5
Chapter 5Factoring
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Factor 2x3 + 6x2y + x + 3y.
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Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 14: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/14.jpg)
Factor 2x3 + 6x2y + x + 3y.
3 2 = 2 + 6 + + 3x x y x y
2 = 2 + 3 + 1 + 3x x y x y
2 = + 3 2 + 1x y x
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Section 5.2
Quadratic Equations
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OBJECTIVES
A Factor trinomials of the form x 2 + bx + c.
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RULEFactoring Rule 1
X 2 + (A + B)X + AB = (X + A)(X + B)
(F1)
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PROCEDUREFactoring x2 + bx + cFind two integers whose product is c and whose sum is b.
1. If b and c are positive, both integers must be positive.
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PROCEDUREFactoring x2 + bx + cFind two integers whose product is c and whose sum is b.
2. If c is positive and b is negative, both integers must be negative.
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PROCEDUREFactoring x2 + bx + cFind two integers whose product is c and whose sum is b.
3. If c is negative, one integer must be negative and one positive.
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Section 5.2Exercise #6
Chapter 5Factoring
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Factor x2 – 8x + 12.
![Page 23: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/23.jpg)
Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 24: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/24.jpg)
Factor x2 – 8x + 12.
= – 6 – 2x x
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Section 5.3
Quadratic Equations
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OBJECTIVES
A Use the ac test to determine whether ax 2 + bx + c is factorable.
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OBJECTIVES
B
Factor ax2 + bx + c by grouping.
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OBJECTIVES
C
Factor ax2 + bx + c using FOIL.
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TEST
ac test for ax2 + bx + c
A trinomial of the form ax2 + bx + c is factorable if there are two integers with product ac and sum b.
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TESTac test
ax2 + bx + c
We need two numbers whose product is ac.
The sum of the numbers must be b.
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PROCEDUREFactoring by FOIL
ax2 + bx + c = (__x + __)(__x + __)
Product must be c.
Product must be a.
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PROCEDURE
1. The product of the numbers in the first (F) blanks must be a.
Factoring by FOIL
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PROCEDUREFactoring by FOIL
2. The coefficients of the outside (O) products and the inside (I) products must add up to b.
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PROCEDURE
3. The product of numbers in the last (L) blanks must be c.
Factoring by FOIL
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Section 5.3Exercise #8
Chapter 5Factoring
![Page 36: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/36.jpg)
Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 37: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/37.jpg)
Factor 6x2 – 11xy + 3y2 .
3 2 x x – y – 3y
![Page 38: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/38.jpg)
Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 39: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/39.jpg)
Section 5.4
Quadratic Equations
![Page 40: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/40.jpg)
OBJECTIVES
A Recognize the square of a binomial (a perfect square trinomial).
![Page 41: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/41.jpg)
OBJECTIVES
B Factor a perfect square trinomial.
![Page 42: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/42.jpg)
OBJECTIVES
C Factor the difference of two squares.
![Page 43: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/43.jpg)
RULESFactoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS
X 2 + 2AX + A2 = (X + A)2 (F2)
22 2Note that + ( + )X A X A
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RULESFactoring Rules 2 and 3: PERFECT SQUARE TRINOMIALS
X 2 – 2AX + A2 = (X – A)2
22 2Note that – ( – )X A X A (F3)
![Page 45: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/45.jpg)
RULEFactoring Rule 4: THE DIFFERENCE OF TWO SQUARES
X2– A2 = (X + A)(X – A) (F4)
![Page 46: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/46.jpg)
Section 5.4Exercise #11
Chapter 5Factoring
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Factor 9x2 – 12xy + 4y2 .
![Page 48: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/48.jpg)
Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 49: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/49.jpg)
Factor 9x2 – 12xy + 4y2 .
= 3x – 2y 2
![Page 50: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/50.jpg)
Section 5.4Exercise #13
Chapter 5Factoring
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Factor 16x2 – 25y2 .
![Page 52: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/52.jpg)
Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 53: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/53.jpg)
Factor 16x2 – 25y2 .
= 4 + 5 4 – 5x y x y
![Page 54: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/54.jpg)
Section 5.5
Quadratic Equations
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OBJECTIVES
A Factor the sum or difference of two cubes.
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OBJECTIVES
B Factor a polynomial by using the general factoring strategy.
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OBJECTIVES
C Factor expressions whose leading coefficient is –1.
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RULEFactoring Rule 5: THE SUM OF TWO CUBES.
X 3+ A3 = (X + A)(X 2 – AX + A2)
(F5)
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RULEFactoring Rule 6: THE DIFFERENCE OF TWO CUBES. (F6)
X 3– A3 = (X – A)(X 2+ AX + A2)
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PROCEDUREGeneral Factoring Strategy
1. Factor out all common factors.
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PROCEDUREGeneral Factoring Strategy
2. Look at the number of terms inside the parentheses. If there are:
Four terms: Factor by grouping.
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PROCEDUREGeneral Factoring Strategy
Three terms:
If the expression is a perfect square trinomial, factor it. Otherwise, use the ac test to factor.
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PROCEDUREGeneral Factoring Strategy
Two terms and squared:
Look at the difference of two squares (X
2–A2) and factor it.
Note: X 2+A2 is not factorable.
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PROCEDUREGeneral Factoring Strategy
Two terms and cubed:
Look for the sum of two cubes (X
3+A3) or the difference of two cubes (X
3-A3) and factor it.
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PROCEDUREGeneral Factoring Strategy
Make sure your expression is completely factored.
Check by multiplying the factors you obtain.
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Chapter 5Factoring
Section 5.5
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Section 5.5Exercise #15
Chapter 5Factoring
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Factor 8y3 – 125x3 .
3 3 = 2 – 5y x
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Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
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Factor 8y3 – 125x3 .
= 2y 3 – 5x 3
= 2y – 5x 4y2 + 10xy + 25x2
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Section 5.5Exercise #17
Chapter 5Factoring
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Factor 2x3 – 8x2 – 10x.
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Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 74: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/74.jpg)
Factor 2x3 – 8x2 – 10x.
2 = 2 – 4 – 5x x x
= 2 – 5 + 1x x x
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Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 76: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/76.jpg)
Factor 2x3 + 6x2 + x + 3.
2 = + 3 2 + 1x x
2 = 2 + 3 + 1 + 3x x x
3 2 = 2 + 6 + + 3x x x
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Section 5.5Exercise #20
Chapter 5Factoring
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Factor – 9x4 + 36x2 .
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Factor out GCF
(3) Terms(2) Terms (4) Terms
Difference of Squares
Sum/Difference of Cubes
Perfect Square Trinomial
(x2 + bx + c)
(ax2 + bx + c)
Grouping
Factoring Strategy Flow Chart
![Page 80: Section 5.1 Quadratic Equations. OBJECTIVES A Find the greatest common factor (GCF) of numbers. B Find the GCF of terms.](https://reader030.fdocuments.in/reader030/viewer/2022032703/56649cf95503460f949ca51d/html5/thumbnails/80.jpg)
Factor – 9x4 + 36x2 .
2 2 = – 9 – 4x x
2 = – 9 + 2 – 2x x x
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Section 5.6
Quadratic Equations
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OBJECTIVES
A Solve quadratic equations by factoring.
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DEFINITIONQuadratic Equation in Standard Form
If , and are real numbers (a 0),
a b c
ax2+ bx + c = 0
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PROCEDURESolving Quadratics by Factoring
1. Perform necessary operations on both sides so that right side = 0.
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PROCEDURESolving Quadratics by Factoring
2. Use general factoring strategy to factor the left side if necessary.
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PROCEDURESolving Quadratics by Factoring
3. Use the principle of zero product and make each factor on the left equal 0.
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PROCEDURESolving Quadratics by Factoring
4. Solve each of the resulting equations.
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PROCEDURESolving Quadratics by Factoring
5. Check results by substituting solutions obtained in step 4 in original equation.
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Section 5.6Exercise #24
Chapter 5Factoring
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(2x – 3)(x – 4) = 2(x – 1) – 1Solve.
2x2 – 8x – 3x + 12 = 2x – 2 – 1
2x2 – 11x + 12 = 2x – 3
2x2 – 13x + 12 = – 3
2x2 – 13x + 15 = 0
2x – 3 = 0 x – 5 = 0or
(2x – 3)(x – 5) = 0
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(2x – 3)(x – 4) = 2(x – 1) – 1Solve.
2x – 3 = 0 x – 5 = 0or
2x = 3 x = 5
x =
32
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Section 5.7
Quadratic Equations
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OBJECTIVESA Integer problems.
B Area and perimeter problems.
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OBJECTIVESC Problems involving the
Pythagorean Theorem.
D Motion problems.
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NOTETerminology
Examples: 3,4; – 6,–5
Notation2 consecutive integers
n, n+1
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NOTETerminology
Examples: 7, 8, 9; – 4,– 3,– 2
Notation3 consecutive integers
n, n+1, n+2
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NOTETerminology
Examples: 8,10; – 6,– 4
Notation2 consecutive even integers
n, n +2
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NOTETerminology
Examples: 13,15; – 21,– 19
Notation2 consecutive odd integers
n, n +2
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DEFINITIONPythagorean Theorem
a2 + b2 = c2
If the longest side of a right triangle is of length c and the other two sides are of length a and b, then
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DEFINITIONPythagorean Theorem
a2 + b2 = c2
Leg a
Hypotenuse c
Leg b
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Section 5.7Exercise #26
Chapter 5Factoring
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The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.
Let x + 2 = 2nd odd integer
x (x + 2) = 10(x + 2) + 13
x2 + 2x = 10x + 20 + 13
x2 + 2x = 10x + 33
x2 – 8x = 33
x2 – 8x – 33 = 0
Let x = 1st odd integer
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The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.
x2 – 8x – 33 = 0
(x – 11)(x + 3) = 0
x – 11 = 0 x + 3 = 0 or
x = 11 x = – 3 or
Let x = 1st odd integer
Let x + 2 = 2nd odd integer
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The product of two consecutive odd integers is 13 more than 10 times the larger of the two integers. Find the integers.
x = 11 x = – 3 or
x + 2 = 13 x + 2 = – 1
The integers are 11 and 13 or – 3 and – 1.
Let x = 1st odd integer
Let x + 2 = 2nd odd integer
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Section 5.7Exercise #29
Chapter 5Factoring
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A rectangular 10-inch television screen (measured diagonally) is 2 inches wider than it is high. What are the dimensions of the screen?
x2 + (x + 2)2 = 102
Let x = height
x
Let x + 2 = length
x + 2
10
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x2 + (x + 2)2 = 102
x 2 + x2 + 4x + 4 = 100
2x2 + 4x + 4 = 100
2x2 + 4x – 96 = 0
x2 + 2x – 48 = 0
(x + 8)(x – 6) = 0
x + 8 = 0 x – 6 = 0 or x = – 8 x = 6
height –8 x + 2 = 8
The screen is 6 inches high and 8 inches long.