quadratic functions and equations - ddtwo
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Quadratic Functions and Equations
What is Quadratic Function? Equation?
How to Solve Quadratic Function/ Equation?
Friday, January 31, 2020
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Quadratic Function and Equation
A quadratic function is in the standard form y = ax2 + bx + c or f(x) = ax2 + bx + c
A quadratic equation is a quadratic function equated to zero
The standard quadratic equation form is ax2 + bx + c = 0 where a, b, and c are numbers with a 0.
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Solving Quadratic Equation
Quadratic Equation can be solve by
Taking the square root
Factoring
Completing the square
Quadratic formula
Graphing calculator
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Solving Quadratic Equation
Only Quadratic Equation of the form
ax2 + c = 0
Example:
Which of the following quadratic equation can be solved by taking the square root?
can be solved by taking the square root.
B.) 3x2 – 9x = 0A.) 2x2 + 8 = 0
C.) x2 + 4x – 5 = 0 D.) none of these
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Solving Quadratic Equation
Example: Solve by taking the square root:
1.) 2x2 – 8 = 0 2.) (x – 3)2 + 8 = 44
2x2 = 8
2 2
4
x2 = 4
x2 =
x = 2
x = 2 Or x = -2
(x – 3)2 = 36
(x – 3)2 = 36
x – 3 = 6
x = 6 + 3 Or x = -6 + 3
x = 9 Or x = -3
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Solving Quadratic Equation
Example: Solve by taking the square root:
3.) 2x2 + 2 = 0 4.) (x – 3)2 + 8 = -28
2x2 = -2
2 2
-1
x2 = -1
x2 =
x = i
x = i Or x = -i
(x – 3)2 = -36
(x – 3)2 = -36
x – 3 = 6i
x = 3 + 6i Or x = 3 – 6i
Numbers with i is called an imaginary number
Real and Imaginary together are called Complex number
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Solving Quadratic Equation
Try this! : Solve by taking the square root:
5.) x2 – 8 = 0 6.) (x + 1)2 + 8 = 58
x2 = 8
8x2 =
x = 2
x = 2
Or x = -2
(x + 1)2 = 50
(x + 1)2 = 50
x + 1 = 5
x = -1 + 5
Or x = -1 – 5
2
2
2
2
2
2
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Solving Q. E. by Factoring
Example: Factor and use the zero property to solve the Q.E.
1. x² + 3x − 10 = 0
= 0( x ) ( x ) 2 and -5 -3
-2 and 5 3
1 and -10 -9
-1 and 10 9
- 2 + 5
Factors of -10 Sum of Factors
Using zero property
x – 2 = 0 or x + 5 = 0
x = 2 or x = -5
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Solving Q. E. by Factoring
Example: Factor and use the zero property to solve the Q.E.
2. x² − 5x + 6 = 0
= 0( x ) ( x ) 2 and 3 5
1 and 6 7
-2 and -3 -5
-1 and -6 -7
- 2 - 3
Factors of 6 Sum of Factors
Using zero property
x – 2 = 0 or x – 3 = 0
x = 2 or x = 3
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Solving Q. E. by Factoring
Example: Factor and use the zero property to solve the Q.E.
3. x² − 2x − 3 = 0
= 0( x ) ( x ) 1 and -3 -2
-1 and 3 2+ 1 - 3
Factors of -3 Sum of Factors
Using zero property
x + 1 = 0 or x – 3 = 0
x = -1 or x = 3
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Solving Q. E. by Factoring
Example: Box Method Factoring and the zero property to solve the Q.E.
F of -18 Sum
-9, 2 -7
9,-2 7
-6,3 -3
6,-3 3
1,-18 -17
-1,18 17
1. 3x2 + 7x – 6 = 0
1st: (3)(-6) = -18
2nd:3rd: Box Method
3x2
-6
9x
-2x
x 3
3x
-2
(x + 3)(3x – 2) = 0
GCF
x + 3 = 03x – 2 = 0+2 +2
3x = 23 3
x 23
=
x =-3
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Solving Q. E. by Factoring
F of -168 Sum
-21, 8 -13
21,-8 13
-12,14 2
12,-14 -2
28,-6 22
-28,6 17
2. 8x2 + 22x – 21 = 0
1st: (8)(-21) = -168
2nd:3rd: Box Method
8x2
-21
28x
-6x
2x 7
4x
-3
(2x + 7) (4x – 3) = 0
GCF
Example: Box Method Factoring and the zero property to solve the Q.E.
4x – 3 = 0+3 +3
4x = 34 4
x 34
=
2x + 7 = 0- 7 -7
2x = -72 2
x -72
=
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Solving Quadratic Equation
Example: Solve by factoring
1.) 6x2 + 7x – 5 = 0
2.) x2 – 6x = 27
(2x - 1)(3x + 5) = 0
(2x - 1) = 0 Or (3x + 5) = 0
x2 – 6x – 27 = 0
(x + 3)(x – 9) = 0
(x + 3) = 0 Or (x – 9) = 0
x = -3 Or x = 9
x = 1/2 Or x = -5/3
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Solving Quadratic Equation
Solving by Completing the Square
Recall: Perfect Square Trinomials
Examples
x2 + 6x + 9
x2 - 10x + 25
x2 + 12x + 36
= (x + 3)(x + 3)
= (x + 3)2
= (x + 6)(x + 6)
= (x – 5)(x – 5)
= (x – 5)2
= (x + 6)2
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Creating a Perfect Square Trinomial
In the following perfect square trinomial, the constant term is missing. X2 + 14x + ____
Find the constant term by squaring half the coefficient of the linear term (the number beside x).
(14/2)2
X2 + 14x + 49
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Creating a Perfect Square Trinomial
Create a perfect square trinomial and factor
x2 + 20x + ___
x2 - 4x + ___
x2 + 5x + ___
100
4
25/4
= (x + 10)2
= (x – 2)2
= (x + 5/2)2
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Solving Quadratic Equation
Solving by Completing the Square
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Solving Quadratic Equation
Solve by Completing the Square Example 1
Step 1: Move quadratic term, and linear term to left side and the constant term to right side of the equation
x2 + 8x – 20 = 0
+ 20 +20
x2 + 8x = 20Step 2: Find the number that completes the square on the left side and add to both sides.of equation
x2 + 8x = 20+ 16 +16
Step 3: Factor the left side of the equation and simplify the right side of the equation
(x + 4)2 = 36
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Solve by Completing the Square
2( 4) 36x
( 4) 6x
Solving Quadratic Equation
Step 4: Solve by taking the square root
4 6
4 6 an
d 4 6
10 and 2 x=
x
x x
x
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Solve by Completing the Square Example 2
22 7 12 0x x
22 7 12x x
Solving Quadratic Equation
Step 1: Move quadratic term, and linear term to left side and the constant term to right side of the equation
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Step 2:
Find the term that completes the square on the left side of the equation. Add that term to both sides.
2
2
2
2 7
2
2 2 2
7 12
7
2
=-12 +
6
x x
x x
xx
21 7 7 49
( ) then square it, 2 62 4 4 1
7
2 49 49
16 1
76
2 6x x
Solving Quadratic Equation
The quadratic coefficient must be equal to 1 before you complete the square, so you must divide all terms by the quadratic coefficient first.
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Step 3:
Factor the perfect square trinomial on the left side of the equation. Simplify the right side of the equation.
2
2
2
76
2
7 96 49
4 16 16
7 47
4
49 49
16 1
16
6x x
x
x
Solving Quadratic Equation
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27 47( )
4 16x
7 47( )
4 4
7 47
4 4
7 47
4
x
ix
ix
Solving Quadratic Equation
Step 4: Solve by taking the square root
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2
2
2
2
2
1. 2 63 0
2. 8 84 0
3. 5 24 0
4. 7 13 0
5. 3 5 6 0
x x
x x
x x
x x
x x
Try the following examples. Do your work on your paper and then check your answers.
1. 9,7
2.(6, 14)
3. 3,8
7 34.
2
5 475.
6
i
i
Solving Quadratic Equation
Solve by Completing the Square
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Solving Quadratic Equation
Solving by Quadratic Formula
ax2 + bx + c = 0
ax2 + bx = -c
a
cx
a
bx2
2
2
2
22
a4
b
a
c
a4
b x
a
bx
The quadratic formula is derived by completing the square using the standard quadratic equation:
2
22
a4
ac4b
a2
bx
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Solving Quadratic Equation
Solving by Quadratic Formula
a2
ac4bbx
2
The Quadratic Formula is
The standard quadratic equation form is ax2 + bx + c = 0 where a, b, and c are numbers with a 0.
All quadratic equation can be solved by using the quadratic formula
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Solving Quadratic Equation
Solve by quadratic formula
1.) 2x2 + 4x = 5 2x2 + 4x – 5 = 0
a2
ac4bbx
2
a=2 ,b=4, and c=-5
)2(2
)5)(2(4)4(4 2 x
4
40164x
4
564x
4
1424x
2
142x
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Solving Quadratic Equation
Solve by quadratic formula
2.) x2 + 13 = 4x x2 – 4x + 13 = 0
a2
ac4bbx
2
a=1 ,b=-4, and c=13
)1(2
)13)(1(4)4()4(x
2
2
52164x
2
364x
2
i64x
i32x
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Changing equation in Standard
Change the equation in standard quadratic form and identify a,b, and c then solve using Quadratic Formula
1.) 2x2 + 8 = 0
4.) (x – 3)2 + 8 = 44
2.) 3x2 = 9x
3.) x2 + 4x = 5
a=2 ,b=0, and c=8
3x2 – 9x = 0
2x2 + 8 = 0
a=3 ,b=-9, and c=0
x2 + 4x – 5 = 0
a=1 ,b=4 and c=-5
x2 – 6x – 27 = 0
a=1 ,b=-6, and c=-27
ax2 + bx + c = 0Standard form
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Solving Quadratic Equation
Solve by quadratic formula
1.) x2 – 3x – 2 = 0
2.) 2x2 + 13 = 8x
3.) 3x2 – 3 = 4x
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Find Quadratic Equation
Finding a Quadratic Function or Equation given its Roots, Zeros or Solution.
Example:
1. Find a Quadratic function/ equation whose zeros are x = 1 and x = -3
Solution:
f(x) =
(x – 1) (x + 3)
f(x) = x2 + 3x - 1x - 3
f(x) = x2 + 2x - 3 or x2 + 2x – 3 = 0
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Find Quadratic Equation
Finding a Quadratic Function or Equation given its Roots, Zeros or Solution.
Try it yourself:
2. Find a Quadratic function/ equation whose zeros are x = 2 and x = -1
Solution:
f(x) =
(x – 2) (x + 1)
f(x) = x2 + 1x - 2x - 2
f(x) = x2 - 1x - 2 or x2 - 1x – 2 = 0