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


• 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


• Example:

1,5.2 :solutions

01or 052

0)1)(52( :factored

0572 :form standard

7522

2

xx

xx

xx

xx

xx


• 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


• Example:

5

32or

5

32

325or 325

325or 325

325 2

xx

xx

xx

x


• Example of completing the square:

2323

2)3(02)3(

square) the(complete 0296

factored becannot 076

22

2

2

xx

xx

xx

xx


• 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


• Review:

• x4 + y4 – can be factored by completing the square

))((

))((

(prime)

))((

2233

2233

22

22

yxyxyxyx

yxyxyxyx

yx

yxyxyx

))()(())(( 22222244 yxyxyxyxyxyx


• 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


• 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


• 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


• 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


• 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


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


• 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


• 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)(


• 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


Vertex = (h, k)

khxxf 2)(


• Vertex Formula:The graph of f(x) = ax2 + bx + c has vertex

a

bf

a

b

2,

2


• 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


• Proportion – statement that two ratios are equal:

Solve using cross multiplication:

dc

ba

bcad


• Solve for x:

Solution:

79

381 x

60

9540

279567

)3(9781

x

x

x

x


• 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


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


• 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


• Conversions

mlteaspoon

quartsliter

mlcc

cccmliter

51

057.11

11

100010001 3


• Try these:

– 3.5 liters = ________ ml

– liter = ________ cc81

7.1 quadratic equations quadratic equation: zero-factor property: if a and b are real numbers and if...

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