Download forms of quadratic equations

FORMS OF QUADRATIC EQUATIONS
TOPIC
PAGE
General form to transformational form (when a = 1)
4
General form to transformational form
8
(when a = 1)
Transformational form to standard form
14
Standard form to transformational form
18
Standard form to general form
22
© 2003 Paul Lobel
Careerwise Tutoring
Easy Math Steps
722-3733
1
YOU SHOULD KNOW
GENERAL FORM OF A QUADRATIC EQUATION
y  ax  bx  c
2
ANY NUMBER
ANY NUMBER
EXCEPT ZERO
© 2003 Paul Lobel
Careerwise Tutoring
ANY NUMBER
General form
y  2x 2  4x  13
Standard form
y  2(x  2)2 - 15
Transformational form
1
y  15  x  22
2
Mapping rule
© 2003 Paul Lobel
Careerwise Tutoring
(x, y) - - - - - - - --  (x - 2, 2y - 15)
3
© 2003 Paul Lobel
Careerwise Tutoring
4
EXAMPLE
CHANGING FROM TRANSFORMATIONAL FORM TO
STANDARD FORM
Quadratic equation in
transformational form
1
2
 y  2  x  1
3
© 2003 Paul Lobel
Careerwise Tutoring
STEP ONE
STEP TWO
Change the number in front of the left
Change the number after the
bracket to its reciprocal and move it to
y value to its opposite and move it
the front of the right bracket
to the right hand side.
y  2 
- 3x  1
2
y  - 3x  1 - 2
2
5
PRACTICE QUESTIONS
CHANGING FROM TRANSFORMATIONAL FORM TO
STANDARD FORM
Quadratic equation in
transformational form
STEP ONE
STEP TWO
Change the number in front of the left
Change the number after the
bracket to its reciprocal and move it to
y value to its opposite and move it
the front of the right bracket
to the right hand side.
 2y  2  x  1
2
2y  3  x  12
2

2
y  7   x  82
3
4y  4  x  4
2
1
© 2003 Paul Lobel
Careerwise Tutoring
6
ANSWERS TO PRACTICE QUESTIONS
CHANGING FROM TRANSFORMATIONAL FORM TO
STANDARD FORM
Quadratic equation in
transformational form
 2y  2  x  1
2
2y  3  x  12
2
STEP ONE
STEP TWO
Change the number in front of the left
Change the number after the
bracket to its reciprocal and move it to
y value to its opposite and move it
the front of the right bracket
to the right hand side.
y -
y  3  1 x  122
y
y  7   - 3 x  82
2
y -
y  4  1 x  42
y
2

2
y  7   x  82
3
4y  4  x  4
2
© 2003 Paul Lobel
Careerwise Tutoring
1
x  12 - 2
2
y  2  - 1 x  12
2
4
1
x  122  3
2
3
x  82 - 7
2
1
x  42 - 4
4
7
© 2003 Paul Lobel
Careerwise Tutoring
8
EXAMPLE
CHANGING FROM STANDARD FORM TO
TRANSFORMATIONAL FORM
Quadratic equation in
standard form
y  - 3x  1 - 2
2
© 2003 Paul Lobel
Careerwise Tutoring
STEP ONE
STEP TWO
Change the number after the right
Change the number in front of the
bracket to its opposite and move it to
right bracket to its reciprocal and
the left side.
move it in front of the left bracket.
y  2  - 3x  1
2

1
y  2  x  12
3
9
PRACTICE QUESTIONS
CHANGING FROM STANDARD FORM TO
TRANSFORMATIONAL FORM
Quadratic equation in
standard form
STEP ONE
STEP TWO
Change the number after the right
Change the number in front of the
bracket to its opposite and move it to
right bracket to its reciprocal and
the left side.
move it in front of the left bracket.
y  - 7x  1 - 3
2
y  4x  9  6
2
y -
1
x  32 - 7
3
y  - 0.5x  3  4
2
© 2003 Paul Lobel
Careerwise Tutoring
2
10
ANSWERS TO PRACTICE QUESTIONS
CHANGING FROM STANDARD FORM TO
TRANSFORMATIONAL FORM
Quadratic equation in
standard form
STEP ONE
STEP TWO
Change the number after the right
Change the number in front of the
bracket to its opposite and move it to
right bracket to its reciprocal and
the left side.
move it in front of the left bracket.
y  - 7x  1 - 3
y  3 
y  4x  9  6
y  6  4x  9
2
2
1
2
y  - x  3 - 7
3
y  - 0.5x  3  4
2
© 2003 Paul Lobel
Careerwise Tutoring
- 7x  1
2
2
y  7  
y  4 
-
-
1
x  32
3
1
x  32
2

1
y  3  x  12
7
1
y  6  x  92
4
 3y  7   x  3
2
 2y  4  x  3
2
11
© 2003 Paul Lobel
Careerwise Tutoring
12
EXAMPLE
CHANGING FROM STANDARD FORM TO
GENERAL FORM
STEP ONE
STEP TWO
STEP THREE
Simplify
Quadratic equation in
standard form
FOIL the binomial square to
Multiply number in front of
get a perfect square trinomial
bracket by numbers in bracket
y  - 3x  1 - 6
y  -3x 2  2x  1  6
2
© 2003 Paul Lobel
Careerwise Tutoring
y  -3x 2  6x - 3  6
y  -3x 2  6x-9
13
PRACTICE QUESTIONS
CHANGING FROM STANDARD FORM TO
GENERAL FORM
Quadratic equation
in standard form
STEP ONE
STEP TWO
STEP THREE
FOIL the binomial square to
Multiply number in front
Simplify
get a perfect square
of bracket by numbers in
trinomial
bracket
y  2x  1 - 4
2
y  - 7x  1 - 5
2
y
1
x  62  3
2
y 
1
 x  82 -2
4
© 2003 Paul Lobel
Careerwise Tutoring
3
14
ANSWERS TO PRACTICE QUESTIONS
CHANGING FROM STANDARD FORM TO
GENERAL FORM
Quadratic equation
in standard form
y  2x  1 - 4
2
STEP ONE
STEP TWO
STEP THREE
FOIL the binomial square to
Multiply number in front
Simplify
get a perfect square
of bracket by numbers in
trinomial
bracket
y  2x 2  2x  1  4
y  2x  4x  2  4


2
y  - 7x  1 - 5 y  7 x  2x  1  5 y  -7x  14x - 7  5
2
2
1 2
1
2
x  12x  36  3
y

y  x  6   3
2
2
1
2
y  x  8  - 2
4
© 2003 Paul Lobel
Careerwise Tutoring
y
1 2
x  16x  64  2
4
2
1
y  x 2  6x  18  3
2
1
y  x 2  4x  16 - 2
4
y  2x 2  4x - 2
y  -7x 2  14x - 12
1
y  x 2  6x  21
2
1
y  x 2  4x  14
4
15
© 2003 Paul Lobel
Careerwise Tutoring
16
EXAMPLE
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when a = 1 )
Quadratic equation
in general form
y  x 2  4x  7
STEP ONE
STEP TWO
STEP THREE
Move constant "c"
to opposite side
and change sign.
Add two blank spaces
Find the missing number to
make a trinomial perfect
square and add this number
to both sides.
Simplify the left side and
y  7  ___  x 2  4x  ___
y  7  4  x  4x  4
2
factor the right side.
y  11 
(x  2)2
a=1
© 2003 Paul Lobel
Careerwise Tutoring
17
PRACTICE PROBLEMS
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when a = 1 )
Quadratic equation in
general form
STEP ONE
STEP TWO
STEP THREE
Move constant "c"
to opposite side
and change sign.
Add two blank spaces
Find the missing number
to complete the square and
add it to both sides.
Simplify the left side and
factor the right side.
y  x 2  8x  4
y  x 2  3x  18
y  x 2  10x
y  x2  x  7
© 2003 Paul Lobel
Careerwise Tutoring
18
ANSWERS TO PRACTICE PROBLEMS
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when a = 1 )
STEP ONE
STEP TWO
STEP THREE
Move constant "c"
to opposite side
and change sign.
Add two blank spaces
Find the missing number to
complete the square and
add it to both sides.
Simplify the left side and
y  x 2  8x  4
y  4  ___  x 2  8x  ___
y  4  16  x 2  8x  16
y  x  3x  18
y  18  ___  x 2  3x  ___
Quadratic equation in
general form
2
9
9
y  18   x 2  3x 
4
4
y  x 2  10x
y  ___  x 2  10x  ___
y  25  x 2  10x  25
y  x2  x  7
y  7  ___  x 2  x  ___
1
1
y  7   x2  x 
4
4
© 2003 Paul Lobel
Careerwise Tutoring
factor the right side.
y  20 
(x  4)2
81  
3

y    x  
4 
2

y  25 
2
(x  5)2
27 
1 2

 y    (x  )
4 
2

19
© 2003 Paul Lobel
Careerwise Tutoring
20
EXAMPLE
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when "a" greater than 1 )
Quadratic equation in general form
Move constant to left hand side and change its sign (+7)
Factor the x terms so that x2 has a coefficient of 1.
(Divide both x terms by 2 and put the 2 in front of brackets)
Add blanks spaces to both sides as shown
Find the missing number to make a perfect square trinomial
in the right bracket. (4)
Multiply the missing number by the number in front of the
right bracket and add it to the left side. (Add 2 x 4 = 8)
Simplify the left side by adding and change the trinomial to
the square of a binomial (x + 2)2
Enclose the left had side in a bracket. Change the number
in front of the right bracket to its reciprocal and put it in
front
the Careerwise
left bracket.
© 2003
Paul of
Lobel
Tutoring (1/2)
y  2x 2  4x  7
y  7  2x 2  4x
y  7  2 x 2  2x 
y  7  ___  2x 2  2x  ___ 
y  7  ___  2x 2  2x  1
y  7  2  2x 2  2x  1
y  9  2x  1
2
1
y  9  x  12
2
21
PRACTICE PROBLEMS
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when "a" greater than 1 )
Quadratic equation in general form
y  - 3x 2  6x  6
Move constant to left hand side and change its sign
Factor the x terms so that x2 has a coefficient of 1.
Add blanks spaces to both sides as shown
Find the missing number to make a perfect square trinomial
in the right bracket.
Multiply the missing number by the number in front of the
bracket and add it to the left side.
Simplify the left side by adding and factor the trinomial to
the square of a binomial
Enclose the left had side in a bracket. Change the number
in front of the right bracket to its reciprocal and put it in
front
the Careerwise
left bracket.
© 2003
Paul of
Lobel
Tutoring
5
22
ANSWER TO PRACTICE PROBLEM
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when "a" greater than 1 )
Quadratic equation in general form
y  - 3x 2  6x  6
Move constant to left hand side and change its sign
y  6  - 3x 2  6x
Factor the x terms so that x2 has a coefficient of 1.
y  6  - 3 x 2  2x 
Add blanks spaces to both sides as shown
Find the missing number to make a perfect square trinomial
in the right bracket.
Multiply the missing number by the number in front of the
right brackets and add the result to the left side.
Simplify the left side by adding and factor the trinomial to
the square of a binomial
Enclose the left had side in brackets. Change the number in
front of the right brackets to its reciprocal and put it in front
of the
left brackets.
© 2003
Paul Lobel
Careerwise Tutoring
y  6  ___  - 3x 2  2x  ___ 
y  6  ___  - 3x 2  2x  1
y  6  3  - 3x 2  2x  1
y  3  - 3x  1
2

1
y  3  x  12
3
23
PRACTICE PROBLEM
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when "a" is not equal to 1 )
Quadratic equation in general form
y  4x 2  2x  6
Move constant to left hand side and change its sign
Factor the x terms so that x2 has a coefficient of 1.
Add blanks spaces to both sides as shown
Find the missing number to make a perfect square trinomial
in the right bracket.
Multiply the missing number by the number in front of the
right brackets and add it to the left side.
Simplify the left side by adding and factor the trinomial to
the square of a binomial
Enclose the left had side in a bracket. Change the number
in front of the right bracket to its reciprocal and put it in
front
the Careerwise
left bracket.
© 2003
Paul of
Lobel
Tutoring
6
24
ANSWER TO PRACTICE PROBLEM
CHANGING FROM GENERAL FORM TO
TRANSFORMATIONAL FORM
( when "a" is greater than 1 )
Quadratic equation in general form
y  4x 2  2x  6
Move constant to left hand side and change its sign
y  6  4 x 2  2x
Factor the x terms so that x2 has a coefficient of 1.
1 

y  6  4 x 2  x 
2 

Add blanks spaces to both sides as shown
Find the missing number to make a perfect square trinomial
in the right bracket.
Multiply the missing number by the number in front of the
right brackets and add it to the left side.
Simplify the left side by adding and factor the trinomial to
the square of a binomial
Enclose the left had side in a bracket. Change the number
in front of the right bracket to its reciprocal and put it in
front of the Careerwise
left bracket.
© 2003 Paul Lobel
Tutoring
1


y  6  ___  4 x 2  x  ___ 
2


1
1

y  6  ___  4 x 2  x  
2
16 

1
1
1

y  6   4 x 2  x  
4
2
16 

y 
23
1

 4 x 

4
4

2
1
23 
1

y    x 

4
4 
4

2
25