Post on 01-Jan-2016
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Simplifying Quadratics: Factoring
Simplifying Quadratics: Factoring
Simplifying quadratics makes for easier graphing(x+y)2=x2+2xy+y2
(x-y)2=x2-2xy+y2
(2x+3)2=4x2+12x+9You can apply this property to any
expanded quadratic with perfect squares on x and y
Simplifying quadratics makes for easier graphing(x+y)2=x2+2xy+y2
(x-y)2=x2-2xy+y2
(2x+3)2=4x2+12x+9You can apply this property to any
expanded quadratic with perfect squares on x and y
Simplifying Quadratics: Factoring
Simplifying Quadratics: Factoring
FOILFirst Outer Inner Last
(2x+4)(x-3)2x•x (First)2x•-3 (Outer)4•x (Inner)4•-3 (Last)2x2-6x+4x-12Simplify: 2x2-2x-12
FOILFirst Outer Inner Last
(2x+4)(x-3)2x•x (First)2x•-3 (Outer)4•x (Inner)4•-3 (Last)2x2-6x+4x-12Simplify: 2x2-2x-12
Simplifying Quadratics: Factoring
Simplifying Quadratics: Factoring
Not all quadratics are easily factorable:3x2+5x+2Set up a gridFind a x c (3 x 2)Find all the factorsof 6 that sum to 5Fill in the gridFind the GCF of each column and rowWrite the expression
(3x+2)(x+1)
Not all quadratics are easily factorable:3x2+5x+2Set up a gridFind a x c (3 x 2)Find all the factorsof 6 that sum to 5Fill in the gridFind the GCF of each column and rowWrite the expression
(3x+2)(x+1)
3x2
2
3x
2
X 1
3x
2x
Graphing: TranslationsGraphing: Translations
In ax2+bx+c, c refers to the y-interceptx2+4 is 4 units up
from x2
y=x2+k moves a graph up k units
y=x2-k also moves a graph down k units
In ax2+bx+c, c refers to the y-interceptx2+4 is 4 units up
from x2
y=x2+k moves a graph up k units
y=x2-k also moves a graph down k units
0
1
2
3
45
6
7
8
9
-4 -2 0 2 40
1
2
3
45
6
7
8
9
-4 -2 0 2 4
Graphing: TranslationsGraphing: Translations
In y=(x-h)2
h is the distance moved left or right
-h moves right+h moves left
(x+3)2 is 3 units left of x2
x=h is the line of symmetry
In y=(x-h)2
h is the distance moved left or right
-h moves right+h moves left
(x+3)2 is 3 units left of x2
x=h is the line of symmetry
0
2
4
6
810
12
14
16
18
-4 -2 0 20
2
4
6
810
12
14
16
18
-4 -2 0 2
Graphing: TranslationsGraphing: Translations
In ax2+bx+c:+a opens up-a opens down
To find the line of symmetry:Find -b/2a
In ax2+bx+c:+a opens up-a opens down
To find the line of symmetry:Find -b/2a
0
0.5
1
1.5
22.5
3
3.5
4
4.5
-4 -2 0 2 40
0.5
1
1.5
22.5
3
3.5
4
4.5
-4 -2 0 2 4
Solving Quadratics: Factoring
Solving Quadratics: Factoring To solve by factoring, make ax2+bx+c equal to 0:
0= 3x2+5x+2 Then factor:
0=(3x+2)(x+1) Make each binomial equal to zero and solve:
0=3x+2 0=x+1-2=3x -1=x (second solution)-2/3=x (first solution)
The two solutions are: -2/3 and -1 Solutions to quadratics refer to the x-intercepts
In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it
To solve by factoring, make ax2+bx+c equal to 0:0= 3x2+5x+2
Then factor:0=(3x+2)(x+1)
Make each binomial equal to zero and solve:0=3x+2 0=x+1-2=3x -1=x (second solution)-2/3=x (first solution)
The two solutions are: -2/3 and -1 Solutions to quadratics refer to the x-intercepts
In other words, if the equation is not equal to 0, then we have to make it equal to 0 to solve it
Solving Quadratics: Completing the Square
Solving Quadratics: Completing the Square
X2+10=39 Draw a square with
area x2
Add a rectangle of length 10
Split the rectangle Find the area
Add the area to 39 Find the square root of
the answer (64) Solve x+5=8 and
x+5=-8 X=3 and -13
X2+10=39 Draw a square with
area x2
Add a rectangle of length 10
Split the rectangle Find the area
Add the area to 39 Find the square root of
the answer (64) Solve x+5=8 and
x+5=-8 X=3 and -13
x
x
10
=39
x
x
5
5 25
=39+25
=64+
+
Graphing: Quadratic modeling
Graphing: Quadratic modeling
Coordinates:(0,5), (1, 10), (2,
19)
Use a basic formula:
5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+c
Coordinates:(0,5), (1, 10), (2,
19)
Use a basic formula:
5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+c
02468
101214161820
0 1 2 302468
101214161820
0 1 2 3
Graphing: Quadratic modeling
Graphing: Quadratic modeling
Set up and solve a system for your three equations:
5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+ca=2; b=3; c=5So y=2x2+3x+5
Set up and solve a system for your three equations:
5=a(0)2+b(0)+c10=a(1)2+b(1)+c19=a(2)2+b(2)+ca=2; b=3; c=5So y=2x2+3x+5
Graphing: Quadratic modeling
Graphing: Quadratic modeling
A vehicle’s braking distance is found for the following three coordinates:
Find the equationfor this vehicle’sbraking distance
A vehicle’s braking distance is found for the following three coordinates:
Find the equationfor this vehicle’sbraking distance
Speed (MPH)
Distance (ft.)
0 0
10 19
40 116
Graphing: Quadratic modeling
Graphing: Quadratic modeling
A ball’s trajectory can be found using the equation:
h(t) = -4.9t2 + vot + ho
The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.
A ball’s trajectory can be found using the equation:
h(t) = -4.9t2 + vot + ho
The ball has an initial velocity of 14 m/sec and was thrown from a height of 30 meters.
Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation
The quadratic formula only works when y=0 in y=ax2+bx+c
So, in 10x2-13x-3=0,
The quadratic formula only works when y=0 in y=ax2+bx+c
So, in 10x2-13x-3=0,
€
x =−b ± b2 − 4ac
2a
€
x =13± −132 − 4(10)(−3)
2(10)
Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation
Simplify: do what is under the radical sign first:
Both add and subtract from 13
Simplify:
X=3/2 and -1/5
Simplify: do what is under the radical sign first:
Both add and subtract from 13
Simplify:
X=3/2 and -1/5
€
x =13± 289
20
€
x =13+17
20
€
x =13−17
20
€
x =30
20
€
x =−4
20
Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation
A cat is dropped from a height of 40 feet. Use the formula h=-16t2+44t+40.
1. When does it hit the ground?2. Does it land on it’s feet?
A cat is dropped from a height of 40 feet. Use the formula h=-16t2+44t+40.
1. When does it hit the ground?2. Does it land on it’s feet?
Solving Quadratics: Quadratic EquationSolving Quadratics: Quadratic Equation
Sometimes we want to solve for equations that do not equal 0. So, make them equal to 0:
A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?
Write an equation: 10=-16t2+147t+5 Get the equation in terms of 0:
0=-16t2+147t-5 Solve for t. T=.03 and 9.15 Which value makes more sense? Why?
Sometimes we want to solve for equations that do not equal 0. So, make them equal to 0:
A baseball is hit from home plate into the outfield. If the ball is hit at a height of 5 ft. and an initial velocity of 147 ft/sec, when will it reach a height of 10 ft?
Write an equation: 10=-16t2+147t+5 Get the equation in terms of 0:
0=-16t2+147t-5 Solve for t. T=.03 and 9.15 Which value makes more sense? Why?
DiscriminantDiscriminantAll quadratics
have two solutions, but not all solutions are real:
Discriminant:b2-4ac>0, two real
solutionsb2-4ac=0, one real
solutionb2-4ac<0, two
complex (imaginary) solutions
All quadratics have two solutions, but not all solutions are real:
Discriminant:b2-4ac>0, two real
solutionsb2-4ac=0, one real
solutionb2-4ac<0, two
complex (imaginary) solutions
0
1
2
3
45
6
7
8
9
-4 -2 0 2 40
1
2
3
45
6
7
8
9
-4 -2 0 2 4
Imaginary NumbersImaginary Numbers
All numbers have square roots, even negative numbers:√4=2, -2√-4=2i
Imaginary number: i…for imaginaryi= √-1, so i2=-1
X2=-100 √x2= √-100X= √100• √-1X=10i and -10i
All numbers have square roots, even negative numbers:√4=2, -2√-4=2i
Imaginary number: i…for imaginaryi= √-1, so i2=-1
X2=-100 √x2= √-100X= √100• √-1X=10i and -10i
(√-25)(2i)√25• √-1(2 √-1)5(2) √-12
10(-1)=-10
Complex NumbersComplex Numbers
Complex numbers: a+bi4+2iThe conjugate (opposite) is 4-2i
(4+2i)+(3+i)Only combine like terms: 2i and i are like
terms7+3i
2i(4-7i)8i-14i2=8i-14(-1)=8i+14
Complex numbers: a+bi4+2iThe conjugate (opposite) is 4-2i
(4+2i)+(3+i)Only combine like terms: 2i and i are like
terms7+3i
2i(4-7i)8i-14i2=8i-14(-1)=8i+14