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Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
Vocabulary: Factor: breaking up the equation into binomials
Solution, zeros, x – intercept : solving the equation for x.
Quadratic Equation standard form is ax2 + bx + c
(a, b, c are real numbers)
Part 1) Factor using GCF Method
1) 6 and 12
2) 6 and 15
3) 8x2 + 2x
4) 3x3y – 6x2y2 – 3xy3
Part 2 – All Trinomials – Factor using KrissKross, state the x-intercepts
1) x2 + 5x + 4
2) x2 – 8x + 12
3) 6x2 + 5xy – 4y2
4) 2x2 + 7xy – 4y2
5) x2 + 2x + 5
6) 6x4 + 11x2 – 10
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
Part 3 – Find the solution using square root
What is different with these equations compared to part 2
equations?
Difference of Square (DOS)
u2 - v2 = (u + v)(u - v)
1) r2 – 9
2) x2 – 25
3) 100 – m2
4) 9x2 – 1
5) 49x2 – 100
6) 3d2 – 48
7) 11x2 – 10
8) 2x2 – 15
Part 4 – Factor by Quadratic formula
What is different with these equations compared to part 2
equations?
Quadratic Formula:
Discriminant
 b  b 2  4ac
x
2a
when given ax2 + bx + c = 0
a, b, c are the coefficients
Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have.
Examples:
1: 3x2 + 8x - 14 = 0
a) a =
b=
c=
b) Disc. =
c) solution =
d) factors:
2: 4x2 + 20x + 25 = 0
a)
a=
b=
c=
b) Disc. =
c) zeros =
d) factors:
3) x2 - 9x = 18
4)
25x2 + 9 = 30x
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
Recall…..
Quadratic Formula:
Discriminant
 b  b 2  4ac
x
when given ax2 + bx + c = 0
2a
a, b, c are the coefficients
Find the Discriminant (b2 – 4ac) to determine number and type of solution you will have.
Determine the type and number of solutions:
1: x2 + 6x + 8 = 0
Practice:
1) 2x2 + 4x - 4 = 0
2: x2 + 6x + 9 = 0
2) 8x2 = 9x – 11
3: x2 + 6x = -10
3) 3x – 5x2 + 1 = 6 – 7x
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: N.CN.A.1
Know there is a complex number i such that i2 = -1, and every complex number has the form a + bi with a and b real.
N.CN.A.2
Use the relation i2 = -1 and the commutative, associative, and distributive properties to add, subtract, and multiply complex numbers
Complex numbers
Can you think of a number to make this equation true?
Imaginary number:
i=
1
i2 = -1
Examples:
-25
-13
Examples:
2) –x2 + 4x = 5
1) 2x2 + 11 = -37
Complex Number Standard Form:
a + bi
(real part + imaginary part)
Operations with complex numbers (+, -, x):
Combine real parts and combine imaginary parts.
1)
(8 - i) + (5 + 4i)
3) (7 - 6i) - (3 - 6i)
5) 6i(4 – 2i)
2) (6 + 7i) + (1 - 3i)
4)
(3 + 7i) - (8 - 2i)
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variableMath- Aids.Com | Dynam ically Created Math Worksheets
Part 6: Parabolas – Graphs of Quadratic Equations
y = ax2 + bx + c these quadratic equations form a
Safari Power Saver
Click to Start Flash Plug-in
 shaped graph called a PARABOLA.
Ex A) y = x2
Step 1: Identify a, b, c
a=
b=
c=
Step 2:
Monday September 22, 2014
If “a” is positive, then parabola opens up  . Vertex is a minimum
If “a” is negative, then parabola opens down  . Vertex is a maximum
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Step 3: Identify vertex (Xv, Yv) - ordered pair of turning point
Xv (x coordinate) =
b
2a
To find Yv (y coordinate), substitute Xv into the equation.
V=
Sketch graph
X
y
State x-intercepts:
More instructions at the botto
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1: y = x2 + x – 12
a) a =
b=
c=
b) which way will the graph open? Min or max?
c) vertex:
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d) sketch graph:
X
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y
More instructions at the bottom
State x-intercepts:
2: y = 2x2
a)
a=
b=
c=
b) which way will the graph open? Min or max?
c) vertex:
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
d) sketch graph
X
y
State x-intercepts:
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
Math- Aids.Com | Dynam ically Created Math Worksheets
3: -x2 - 9x + 8 = 0
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a)
a=
b=
c=
b) which way will the graph open? Min or max?
c) vertex:
d) sketch graph
Monday September 22, 2014
X
y
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State x-intercepts:
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To Print the Worksheet: Right Click on the W
If the Answer Key is Selected, Scr
More instructions at the bottom
4: 2x2 – 8x + 6 = 0
a)
a=
b=
c=
b) which way will the graph open? Min or max?
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
c) vertex:
d) sketch graph
X
y
State x-intercepts:
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
Math- Aids.Com | Dynam ically Created Math Worksheets
Safari Power Saver
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5: x2 – 2x - 1 = 0
a)
a=
b=
c=
b) which way will the graph open? Min or max?
c) vertex:
d) sketch graph
Monday September 22, 2014
Worksheet Viewer P
X
y
State x-intercepts:
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If the Answer Key is Selected, Scr
More instructions at the bottom
6: 3x2 – 18x + 20 = 0
a)
a=
b=
c=
b) which way will the graph open? Min or max?
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
c) vertex:
d) sketch graph
X
y
State x-intercepts:
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coordin
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
Word problems:
1) A ball is thrown, its height after t seconds is given by the function h(t) = 7 + 6t – t2 , where t is the time in
seconds after the object is propelled and h is the object’s height in feet.
a) Draw a picture of the situation.
b) How many seconds after the ball is thrown will the object hit the ground?
c) What is the time for it to reach maximum height?
d) What is the maximum height?
e) What is the height at 2 seconds?
2) Given the area of a patio, sketch and label a picture with a possible expression for the width and length.
a) Area = 3x2 – 15x
b) Area = 9 + 54x2
Is it possible to build the patio?
Yes
No
Explain:
Is it possible to build the patio?
Yes No
Explain:
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Monday September 22, 2014
Worksheet Viewe
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
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NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
If the Answer Key is Selected, Sc
Part 6: Parabolas – Graphs of Quadratic Equations Graph using the calculator.
More instructions at the botto
Ex A) y = x2
Step 1: load the equation into “y =” (remember to clear out all equations in there)
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Step 3: fill in some values for table and state x- intercepts
x
y
Step 4: sketch graph.
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor
1: y = x2 + 1
Step 1: load the equation
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
Math- Aids.Com | Dynam ically Created Math Worksheets
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
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Click to Start Flash Plug-in
Step 3: fill in some values for table and state x- intercepts
x
y
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Worksheet Viewer P
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To Print the Worksheet: Right Click on the W
If the Answer Key is Selected, Scr
More instructions at the bottom
Step 4: sketch graph.
2: y = -x2 + 2
Step 1: load the equation
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Step 3: fill in some values for table and state x- intercepts
x
y
Step 4: sketch graph.
3: y = x2 - 2x
Step 1: load the equation
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Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Safari Power Saver
Click to Start Flash Plug-in
Step 3: fill in some values for table and state x- intercepts
x
y
Monday September 22, 2014
Worksheet Viewe
You May Print or Save This Dynam
To Print the Worksheet: Right Click on the
If the Answer Key is Selected, Sc
More instructions at the botto
Step 4: sketch graph.
4: y = -2x2 – 4x + 7
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor
Step 1: load the equation
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Step 3: fill in some values for table and state x- intercepts
x
y
Step 4: sketch graph.
5: y = -x2 + 3
Step 1: load the equation
Math- Aids.Com | Dynam ically Created Math Worksheets
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Safari Power Saver
Click to Start Flash Plug-in
Step 3: fill in some values for table and state x- intercepts
x
y
Monday September 22, 2014
Worksheet Viewe
You May Print or Save This Dynam
To Print the Worksheet: Right Click on the
If the Answer Key is Selected, Sc
More instructions at the botto
Step 4: sketch graph.
6: y = .33x2 – 2x + 4
Step 1: load the equation
http :/ / www.m ath- aids.com / cgi/ pdf_viewer_4.cgi?script_nam e= graphing_coor
Step 2: Identify vertex (Xv, Yv) - ordered pair of turning point
V(
,
)
is the vertex a min or max?
What is the value of the min or max?
Step 3: fill in some values for table and state x- intercepts
x
y
Step 4: sketch graph.
Algebra 2
Solve Quadratic Equations using Factoring - Notes
Name:_______________________________________
Date:_____________
_______________________________________________________________________________________
NJCCCS: A.REI.4 – Solve Quadratic Equations in one variable
Completing the Square
When solving a Quadratic Equation you need to factor it first. This is a short cut to factoring.
If you have a Perfect Square Trinomial (PST) then:
x2 + 2x + 1 = (x + 1)2 = (x + 1)(x + 1)
How do you create a PST?
Take half the 2nd term and then square it.
1)
x2 + 12x + _____
2)
x2 + 14x + _____
3)
x2 - 20x + _____
Solve the Quadratic Equation.
1)
x2 + 12x + 36 = 0
2)
x2 + 14x + 49 = 0
3)
x2 - 20x + 100 = 0
Solve the Quadratic Equation by completing the square. Goal: Try to make a PST.
Step 1: Is left side x2 + bx? NO - move all numbers to right.
Step 2: Add the number to make a PST to BOTH sides.
Step 3: Factor
Step 4: Take square root of BOTH sides (one = +, one = -)
Step 5: solve
1)
x2 + 2x = 0
2)
x2 + 6x = 16
3)
x2 - 2x – 8 = 0
4)
x2 + 8x + 10 = 0