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Exercise 1 Real numbers

Name………………………………………………………………..No……...

Check in the box following each match type. That the correct number.

No

Number Real numbersNumera

lInteg

ernumber

Negative

number

Rational

number

Irrational

number

1 -82 0. 3

¿

3 −132

4 1255 √3

6 1.417 15

3

8 √4×√2

9 0 .23¿

4¿

10 (√6 )2

Score = ………………………… Examiner ………………………………….. ……./……………./……………..

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Exercise 2 Property of equalityName………………………………………………………………..No……...

Put the properties of equality in each of the following correctly.No.

Number Property of equality

1 8 = 8 Reflexive Property

2 9 = 93 100 = 1004 If 6 = 5 + 1 then 5 +

1 = 6Symmetric Property

5 If 8 = 3 + 5 then 3 + 5 = 8

6 If 10 = 7 + 3 then 7 + 3 = 10

7 If 42 = 16 then 16 = 7 + 9 แล้�ว 42 = 7 + 9

Transitive Property

8 If 52 = 25 then 25 = 20 + 5 แล้�ว 52 = 20 + 5

9 If 72 = 49 then 49 = 40 + 9 แล้�ว 72 = 40 + 9

10

If 4 5 = 20 then (4 5) + 3 = 20 + 3

Addition Property


11

If (2 6) = 12 then (2 6) + 2 = 12 + 2

12

If 3 2 = 6 then (3 2) + 5 = 6 + 5

13

If (4 3) = 12 then (4 3) 5 = 12 5

Multiplication Property

14

If (5 2) = 10 then (5 2) 6 = 10 6

15

If 82 = 4 then

82 4

= 4 6


If a, b, c is numbers then property of equality is


CONC

Exercise 3 Property of equality ConceptName………………………………………………………………..No……...

Students to summarize the content on the properties of equality

If a, b, c is real number 1.a = a ex. 10 = 10 is ………………………………………………………2.if a = b then b = a ex. 6 = 3 + 3 then 3 + 3 = 6 is ……………….3.if a = b แล้ะ b = c then a = c ex. 102 = 100 and 100 = 60 + 40 then

102 =60 + 40 is ………………………………………………………………4.if a = b then a + c = b + c ex. 5 2 = 10 and c = 6 then (2 5) + 6 = 10 + 6 is ………………………………………………………………………………..

5.if a = b then ac = bc ex. 102 = 5 and c

= 4 then (102 ) 4 = 5 4

is ………………………………………………………………………………..6.from 1 - 5 summarize the content on the properties of equality is

6.1……………………………………………………………………………………


6.2……………………………………………………………………………………

6.3……………………………………………………………………………………

6.4……………………………………………………………………………………

6.5……………………………………………………………………………………

Exercise 4 Addition propertyName………………………………………………………………..No……...

Explanation Students to put the answer correctly complete

No.

Word property

1 if 2, 6 R Then 2 + 6 R

Closure Property

2 7 + 3 = 3 + 7 Commutative Property of Addition


3 3 + (5 + 4) = (3 + 5) + 4

associativity

4 Real number 0 so 0 + 3 = 3 = 3 + 0

Additive identity

5 (-7) + 7 = 0 = 7 + (-7)

inverse property of addition

6 6 + 3 = 3 + 67 10 + (2 + 8) = (10 +

2) + 88 if 4, -3 R then 4 +

(-3) R9 0 + 8 = 8 = 8 + 010

(-15) + 15 = 0 = 15 + (-15)

Exercise 5 Multiplication propertyName………………………………………………………………..No……...

Explanation Students to put the answer correctly complete

N Word Property


o.1 if 5, 3 R then 2

5 RClosure Property

2 7 2 = 2 7 Commutative Property of multiply

3 5 (4 3) = (5 4) 3

Associativity

4 1 8 = 8 = 8 1

Multiplicative identity

5 13 3 = 1 = 3

13

inverse property of multiply

6 10 3 = 3 107 if 6, 7 R then 7

6 R8 1 10 = 10 =

10 19 1

5 5 = 1 = 5

15

10

if -2, 7 R then (-2) 7 R

11

15 3 = 3 15

12

7 (2 3) = (7 2) 3

13

1 20 = 20 = 20 1

14

6 (4 5) = (6 4) 5

15

10 5 = 5 10


Concept If a, b, c is real number Multiplication property have 1. . …………………………………………………………………………….. 2. ……………………………………………………………………………… 3. ……………………………………………………………………………… 4. ……………………………………………………………………………… 5. ………………………………………………………………………………


Exercise 6 Use real numbers to solving quadraticName…………………………………………………………No……... Class……Explanation : students solve factorization of polynomials, 1-8 by the following example.

EX. Be factorization of polynomials.1. 3x2 + 62. 4x3 + 8x2 - 12x

Solution 1. 3x2 + 6 = 3(x2 + 2)2. 4x3 + 8x2 - 12x = 4x(x2 + 2x - 3)

1) 3x2 + 6x2 = ……………………………………………………………………….2) 2x2 - x = ……………………………………………………………………….3) 4x3 - 16x2 - 8x = ………………………………………………………………4) 5x3 + 15x2 = ……………………………………………………………………….5) 6x3 - 12x2 - 18x= ………………………………………………………………6) 7x2 - 14x = ……………………………………………………………………….7) 9x4 + 18x3 + 27x2 = ………………………………………………………8) 10x2 - 30x= ……………………………………………………………………….

Concept factorization of polynomials used by. ………………………………………………………………..…………………………………..

………………………………………………………………………………………………….


Exercise 7 Use real numbers to solving

quadratic ax2 + bx + c

Name…………………………………………………………No……... Class……Explanation : students solve factorization of polynomials, ax2 + bx + c when a, b, c is Real number and a 0 by the following example

Ex. Be factorization of polynomials of x2 + 5x + 6

Solution Find two numbers multiply is 6 and sum is 5 2 3 = 6 and 2 + 3 = 5 x2 + 5x + 6 = (x + 2)(x + 3)

1) x2 + 7x + 10 = ……………………………………………………………….2) x2 + 8x + 10 = ……………………………………………………………….3) x2 + 7x + 12 = ………………………………………………………………4) x2 + 9x + 8 = ………………………………………………………………5) x2 - 10x + 24 = ………………………………………………………………6) x2 - 4x - 12 = ……………………………………………………………….7) x2 + 3x + 2 = ………………………………………………………………


8) x2 - x - 6 = ………………………………………………………………

Concept factorization of polynomials used by …………………………………………………………………………………………………………

………………………………………………………………………………………………….

………………………………………………………………………………………………….

Exercise 8 Use real numbers to solving

quadratic Name…………………………………………………………No……... Class……

students solve factorization of polynomials, ax2 + bx + c when a, b, c is Real number and a 0 by the following example

Ex. factorization of polynomials 25x2 + 15x + 2

Solution 1. Find a polynomial second degree polynomials multiply like 25x2 ex. (25x)(x) or (5x)(5x)We can write polynomials is

(25x )(x ) or (5x )(5x )2. Find two numbers to multiple are equal. 2 is (2)(1)

or (-2)(-1)Writing a term of two polynomials in the back of Item 1


(25x + 2)(x + 1) or (5x + 2)(5x + 1)(25x - 2)(x - 1) or (5x - 2)(5x - 1)

3. Find the middle term of the polynomial. From the multiplication of polynomials, each pair in 2 equal to the sum. 15x will be.

10xFrom multiply (5x + 2)(5x + 1) middle term is

15x 5x

polynomials is 25x2 + 15x + 2 = (5x + 2)(5x + 1)

1) 3x2 + 10x + 3 = …………………………………………………………….2) 2x2 + x - 6 = ……………………………………………………………3) 8x2 - 2x - 3 = ……………………………………………………………4) 4x2 + 5x - 9 = ……………………………………………………………5) 3x2 + 4x - 15 = ……………………………………………………………6) 2x2 - x - 1 = ……………………………………………………………

factorization of polynomials 1 - 8

Ex factorization of polynomials x2 + 6x + 9 and x2 - 8x + 16

Solution x2 + 6x + 9 =x2 + 2(3)x + (3)2

= (x + 3)2

x2 - 8x + 16 =x2 - 2(4)x + (4)2

= (x - 4)2


1) x2 + 10x + 25 = ……………………………………………………………….2) x2 - 4x + 4 = ……………………………………………………………….3) x2 + 16x + 64 = ………………………………………………………………4) x2 - 12x + 36 = ………………………………………………………………5) x2 - 14x + 49 = ………………………………………………………………6) x2 + 22x + 121= ……………………………………………………………….7) x2 - 24x + 144 = ……………………………………………………………….8) x2 + 30x + 225= ……………………………………………………………….

Concept Second degree polynomial factorization and factor polynomials have a unique degree. Called. …………………………………………………………………………….

And write a formula is.x2 + 2ax + a2 =

………………………………………………………x2 - 2ax + a2 =

………………………………………………………

Exercise 9 Solving quadratic by perfect square.


Name…………………………………………………………No……... Class……factorization of polynomials

1. 7x2 - 14x + 28 =……………………………………………………………..

2. x2 - 78x + 77 =……………………………………………………………..

3. a2b - 12ab + 27b =……………………………………………………………..

4. 7x2 - 2x – 5 =……………………………………………………………..

5. 13m2 + 21m – 10 =……………………………………………………………..

Students of polynomial factorization is made by Perfect square from 1 to 5.By the following example.

Ex. factorization of polynomials x2 - 6x - 2

Solution x2 – 6x – 2 = (x2 – 6x) – 2 = (x2 – 2(3)x + 32) – 2 – 32 = (x2 – 6x + 9) – 2 – 9= (x – 3)2 – 11 = (x – 3)2 – (√11)2

= (x – 3 + √11)(x – 3 - √11)

1. x2 + 8x – 5 =……………………………………………………………..

2. x2 + 8x + 14 =……………………………………………………………..

3. x2 – 10x + 7=……………………………………………………………..

4. x2 + 7x + 11 =……………………………………………………………..

5. -3x2 + 6x + 4 =……………………………………………………………..


Exercise 10 Solving quadratic by perfect

square. Name…………………………………………………………No……... Class……

Show solution

1. factorization of polynomials 1.1 x2 – 8x + 161.2 x2 + 50x + 125

2. factorization of polynomials by perfect square 2.1 x2 – 6x + 72.2 2x2 + 7x – 3 2.3 x2 + 14x – 1

Solution


Exercise 11 Find answer of quality

Name…………………………………………………………No……... Class……

find the answer of quality

No.

quality answer of quality

1 x2 + 8x + 12 = 02 x2 – 6x = 163 2x2 + 7x + 3 = 04 7x2 + 3x – 4 = 05 4x2 + 8x + 3 = 06 x2 – 2x – 10 = 2x + 117 3x2 + 2x – 3 = 08 2x2 + 3x – 7 = 09 10x2 – 20x – 30 = 0

10 –5x2 – 2x + 17 = 0

show solution in your note book

1. Find answer by factorization1.1 x2 – 8x + 12 = 01.2 5x2 + 13x + 6 = 0

2. Find answer by perfect square 2.1 x2 + 10x + 3 = 02.2 x2 – 6x + 4 = 0

3. Find answer by formula x = -b ±√b2 - 4ac2a

3.1 2x2 = -4x + 13.2 x2 – 4x = 21


Exercise 12 QUADRATIC WORD PROBLEMSName………………………………………………………………………………………No……... Class……- We want a single equation that we can solve for the zeros.- At times, we may need to start with two equations if there are two distinct variables. If this happens, rearrange one equation and substitute it into the other so we are left with a single equation.- Solve the equation by factoring or using the Quadratic

Formula.1. When the square of a certain number is diminished by 9 times the number the result is 36. Find the number.2. A certain number added to its square is 30. Find the number.3. The square of a number exceeds the number by 72. Find the number.4. Find two positive numbers whose ratio is 2:3 and whose product is 600.5. The product of two consecutive odd integers is 99. Find the integers.6. Find two consecutive positive integers such that the square of the first is decreased by 17 equals 4 times the second.7. The ages of three family children can be expressed as consecutive integers. The square of the age of the youngest child is 4 more than eight times the age of the oldest child. Find the ages of the three children.8. Find three consecutive odd integers such that the square of the first increased the product of the other two is 224.9. The sum of the squares of two consecutive integers is 113. Find the integers.10. The sum of the squares of three consecutive integers is 302. Find the integers.11. Two integers differ by 8 and the sum of their squares is 130.12. rectangle is 4 m longer than it is wide, its area is 192 m2. Find the dimensions of the rectangle.13. right triangle has a hypotenuse of 10 m. If one of the sides is 2 m longer than the other, find the dimensions of the triangle.


14. The sum of the squares of three consecutive even integers is 980. Determine the integers15. Two integers differ by 4. Find the integers if the sum of their squares is 250.16. Find three consecutive integers such that the product of the first and the last is one less than five times the middle integer.17. A rectangle is three times as long as it is wide. Its area has measure equal to ten times its width measure plus twenty-five. What is the width?18. Richie walked 15 m diagonally across a rectangular field. He then returned to his starting position along the outside of the field. The total distance he walked was 36 m. What are the dimensions of the field?19. A square picture with sides of length 20 cm is to be mounted centrally upon a square frame. If the area of the border of the frame equals the area of the picture, find the width of the border of the frame to the nearest millimetre.20. A square lawn is surrounded by a walk 1 m wide. The area of the lawn is equal to the area of the walk. Find the length of the side of the lawn to the nearest tenth of a metre.21. A picture 20 cm wide and 10 cm high is to be centrally mounted on a rectangular frame with total area three times the area of the picture. Assuming equal margins for all four sides, find the width of the margin.22. A fast food restaurant determines that each 10¢ increase in the price of a hamburger results in 25 fewer hamburgers sold. The usual price for a hamburger is $2.00 and the restaurant sells an average of 300 hamburgers each day. What price will produce revenue of $637.50?23. George operates his own store, George’s Fashion. A popular style of pants sells for $50. At that price, George sells about 20 pairs of pants a week. Experience has taught George that for every $1 increase in price, he will sell one less pair of pants per week. In order to break even, George needs to produce $650/week. How many pairs of pants does George need to sell to break even?24.A biologist predicts that the deer population, P, in a certain

national park can be modelled by where x is the number of years since 1999.a) According to the model, how many deer were in the park in 1999.


b) Will the deer population ever reach zero?c) In what year will the population reach 1000?25. A t-ball player hits a baseball from a tee. The flight of the ball can be modelled by where h is the height in metres, and t is the time in seconds.a) How high is the tee?b) How long does it take the ball to land?c) When does the ball reach a height of 4.5 m?

Exercise 13 QUADRATIC INEQUALITY Name………………………………………………………………………………………No……... Class……

To solve a QUADRATIC of inequality. or find the answer to inequality will require.Properties are not equal. The following example.

Example 1 Find the answers to the following inequality. by the set.1. 2x + x <10.2. 4x - 4 2x + 4.solution : 2x + 2 < 10. 2x + 2 + (-2) < 10 + (-2). 2x < 8. (2x) < (8). x < 4.Set answer 2x + 2 <10 is {x | x <4}.

2. ∵ 4x – 4 2x + 4 4x – 4 + 4 2x + 4 + 4

4x 2x + 8 4x + (-2x) 2x + (-2x) + 8

2x 8

12 (2x)

12 (8)

x 4 Set answer 4x – 4 2x + 4 is {x | x 4}


Example 2 Solve the following. And show you the line number.1. -x + 4 < -122. -x + 7 4

Solution 1. -x + 4 < 12Plus on both sides of the inequality with a -4

-x + 4 + (-4) < -12 + (-4) -x < -8

x > 8 (multiply both sides by -1) Set of inequalities is the answer. The set of real numbers greater

than 8, or {x | x> 8}.

-1 0 1 2 3 4 5 6 7 8 9 10

2. -x + 7 4Plus on both sides of the inequality with a -7.

-x + 7 + (-7) 4 + (-7) -x -3

x 3 set of inequalities is the answer. The set of real numbers less than

or equal to 3 or {x | x 3}

-3 -2 -1 0 1 2 3 4 5 No.

inequality

1 Find the answers to the following inequality. by the set.

1.1 x + 3 < 61.2 2x + 4 101.3 3x – 7 51.4 -4 + 4x 8

1.1 …………………….1.2 …………………….1.3 …………………….1.4 …………………….

2 Solve the following. And show you the line number.2.1 -3x -62.2 -5x – 1 -112.3 -8x + 6 > -10


2.4 -5 – 5x > -2x - 8

Exercise 14 QUADRATIC INEQUALITY Name………………………………………………………………………………………No……... Class……

Example Solve x2 + x – 6 > 0 and show you the number line.Solution x2 + x – 6 > 0

(x + 3)(x – 2) > 0Consider value of x in the interval (-∞ , -3), (-3, 2) and

(2, ∞ )By choosing the value of x in the interval

Interval x (x + 3)(x – 2) (x + 3)(x – 2)(-∞ , -3) -5 (-2)(-7) = 14 Positive(-3, 2) 1 4(-1) = -4 Negative(2, ∞ ) 4 (7)(2) = 14 Positive

When the value of x in the increase is that (x + 3) (x - 2) is positive orGreater than zero on the x in range (-∞ , -3) and (2, ∞ )


Show the answer

-5 -4 -3 -2 -1 0 1 2 3 4

Solve inequity and show answer by number line1. x2 – 9 < 02. x2 – 4x – 5 < 03. x2 + 6x + 9 > 04. x2 + 4x -45. x2 – 9x < 10

Exercise 15 Interval Name………………………………………………………………………………………No……... Class……

The table shows the different types of interval.The elements as the set of real numbers and a, b R by a

< binterval

symbolic

meaning graph on a number line

example

open

(a, b) {x | a < x < b}

(2, 4)

close

[a, b] {x | a x b}

[3, 6]

Half-open

half-closed,

[a, b)

(a, b]

{x | a x < b}

{x | a < x b}

[2, 5)

(2, 5]

Infin (a, ∞ ) {x | x > a} (3, ∞ )


ite [a, ∞ )

(-∞ , a)

(-∞ , a]

{x | x a}

{x | x < a}

{x | x a}

[3, ∞ )

(-∞ , 5)

(-∞ , 5]

Exercise

No.

interval

meaning

graph on a number line

1 (1, 6)2 [2, 7]3 (3, 6]4 [-3, 2)5 (3, ∞ )6 [3, ∞ )7 (-∞ , 4)8 (-∞ , 6]9 (4, 9)

10 [6, 10]


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