7.1 Quadratic Equations Quadratic Equation: Zero-Factor Property: If a and b are real numbers and if...
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Transcript of 7.1 Quadratic Equations Quadratic Equation: Zero-Factor Property: If a and b are real numbers and if...
7.1 Quadratic Equations
• Quadratic Equation:
• Zero-Factor Property:If a and b are real numbers and if ab=0then either a = 0 or b = 0
02 cbxax
7.1 Quadratic Equations
• Solving a Quadratic Equation by factoring1. Write in standard form – all terms on one side
of equal sign and zero on the other
2. Factor (completely)
3. Set all factors equal to zero and solve the resulting equations
4. (if time available) check your answers in the original equation
7.1 Quadratic Equations
• Example:
1,5.2 :solutions
01or 052
0)1)(52( :factored
0572 :form standard
7522
2
xx
xx
xx
xx
xx
7.1 Quadratic Equations
• If 2 resistors are in series the resistance is 8 ohms and in parallel the resistance is 1.5 ohm. What are the resistances?
2,66,2
0)2)(6(
0128812
5.18
)8(5.1
88
22
yxoryx
xx
xxxx
xx
yx
xy
xyyx
7.2 Completing the Square
• Square Root Property of Equations: If k is a positive number and if a2 = k, then
and the solution set is:
k-aka or
}, { k-k
7.2 Completing the Square
• Example of completing the square:
2323
2)3(02)3(
square) the(complete 0296
factored becannot 076
22
2
2
xx
xx
xx
xx
7.2 Completing the Square
• Completing the Square (ax2 + bx + c = 0):1. Divide by a on both sides
(lead coefficient = 1)
2. Put variables on one side, constants on the other.
3. Complete the square (take ½ of x coefficient and square it – add this number to both sides)
4. Solve by applying the square root property
7.2 Completing the Square
• Review:
• x4 + y4 – can be factored by completing the square
))((
))((
(prime)
))((
2233
2233
22
22
yxyxyxyx
yxyxyxyx
yx
yxyxyx
))()(())(( 22222244 yxyxyxyxyxyx
7.2 Completing the Square
• Example:
Complete the square:
Factor the difference of two squares:
222244 yxyx
2222
22222222
2
22
xyyx
yxyyxx
xyyxxyyx 22 2222
7.3 The Quadratic Formula
• Solving ax2 + bx + c = 0:
Dividing by a:
Subtract c/a:
Completing the square by adding b2/4a2:
02 ac
ab xx
ac
ab xx 2
2
2
2
2
44
2
ab
ac
ab
ab xx
7.3 The Quadratic Formula
• Solving ax2 + bx + c = 0 (continued): Write as a square:
Use square root property:
Quadratic formula:
2
2
4442
2 4
42
2
2
a
acbx
ab
aac
ab
a
acb
a
bx
2
4
2
2
a
acbbx
2
42
7.3 The Quadratic Formula
• Quadratic Formula:
is called the discriminant.If the discriminant is positive, the solutions are realIf the discriminant is negative, the solutions are imaginary
a
acbbx
2
42
acb 42
7.3 The Quadratic Formula
• Example:
2,32
1
2
5
2
24255
)1(2
)6)(1(4)5()5(
6c -5,b 1,a 065
2
2
xx
x
xx
7.3 The Quadratic Formula
• Complex Numbers and the Quadratic FormulaSolve x2 – 2x + 2 = 0
i
ii
x
12
22
2
42
2
42
)1(2
)2)(1(4)2()2( 2
7.3 The Quadratic Formula
Method Advantages Disadvantages
Factoring Fastest method Not always factorable
Square root property
Not always this form
Completing the square
Can always be used
Requires a lot of steps
Quadratic Formula
Can always be used
Slower than factoring
bax 2)( :form
7.4 The Graph of the Quadratic Function
• A quadratic function is a function that can be written in the form:f(x) = ax2 + bx + c
• The graph of a quadratic function is a parabola. The vertex is the lowest point (or highest point if the parabola is inverted
7.4 The Graph of the Quadratic Function
• Vertical Shifts:
The parabola is shifted upward by k units or downward if k < 0. The vertex is (0, k)
• Horizontal shifts:
The parabola is shifted h units to the right if h > 0 or to the left if h < 0. The vertex is at (h, 0)
kxxf 2)(
2)( hxxf
7.4 The Graph of the Quadratic Function
• Horizontal and Vertical shifts:
The parabola is shifted upward by k units or downward if k < 0. The parabola is shifted h units to the right if h > 0 or to the left if h < 0 The vertex is (h, k)
khxxf 2)(
7.4 The Graph of the Quadratic Function
• Graphing:
1. The vertex is (h, k).
2. If a > 0, the parabola opens upward.If a < 0, the parabola opens downward (flipped).
3. The graph is wider (flattened) if
The graph is narrower (stretched) if
khxaxf 2)(
10 a
1a
7.4 The Graph of the Quadratic Function
• Vertex Formula:The graph of f(x) = ax2 + bx + c has vertex
a
bf
a
b
2,
2
7.4 The Graph of the Quadratic Function
• Graphing a Quadratic Function:
1. Find the y-intercept (evaluate f(0))
2. Find the x-intercepts (by solving f(x) = 0)
3. Find the vertex (by using the formula or by completing the square)
4. Complete the graph (plot additional points as needed)
18.1 Ratio and Proportion
• Ratio – quotient of two quantities with the same units
Note: Sometimes the units can be converted to be the same.
ba
18.1 Ratio and Proportion
• Proportion – statement that two ratios are equal:
Solve using cross multiplication:
dc
ba
bcad
18.1 Ratio and Proportion
• Example: E(volts)=I(amperes) R(ohms)How much current for a circuit with 36mV and resistance of 10 ohms?
mAAI
VmVmVI
6.3106.3
106.36.310
36
3
3
18.2 Variation
• Types of variation:1. y varies directly as x:
2. y varies inversely as x:
3. y varies directly as the square of x:
4. y varies directly as the square root of x:
2kxy
kxy
x
ky
xky
18.2 Variation
• Solving a variation problem:1. Write the variation equation.
2. Substitute the initial values and solve for k.
3. Rewrite the variation equation with the value of k from step 2.
4. Solve the problem using this equation.
18.2 Variation• Example: If t varies inversely as s and
t = 3 when s = 5, find s when t = 5
1. Give the equation:
2. Solve for k:
3. Plug in k = 15:
4. When t = 5: 315515
5 sss
s
kt
155
3 kk
st
15
B.1 Introduction to the Metric System
• Metric system base units:
(L)liter volume
(cd) candela intensity luminous
(A) ampere current electrical
(mol) molesubstanceofamount
C)( Celsius degreesetemperatur
(s) secondtime
(kg) kilogramt)mass(weigh
(m)meter length
B.1 Introduction to the Metric System
Multiple in decimal form
Power of 10 Prefix Symbol
1000000 106 mega M
1000 103 kilo k
100 102 hecto h
10 101 deka da
1 100 base unit
0.1 10-1 deci d
0.01 10-2 centi c
0.001 10-3 milli m
0.000001 10-6 micro
B.1 Introduction to the Metric System
• 1 gram = weight of 1 ml of water
• Unit of weight = 1Kg = 1000 grams
• 1 liter of water weighs 1 Kg
B.2 Reductions and Conversions
• Conversions:
Note: In Canada, speed is in kph instead of mph
mileskm
mileskm
inchesm
inchescm
1.35
621.01
37.391
394.01
B.2 Reductions and Conversions
• Conversions
mlteaspoon
quartsliter
mlcc
cccmliter
51
057.11
11
100010001 3