Quadratic Functions and Equations
What is Quadratic Function? Equation?
How to Solve Quadratic Function/ Equation?
Friday, January 31, 2020
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.
Solving Quadratic Equation
Quadratic Equation can be solve by
Taking the square root
Factoring
Completing the square
Quadratic formula
Graphing calculator
Friday, January 31, 2020
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
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
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
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
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
Friday, January 31, 2020
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
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
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
Friday, January 31, 2020
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
=
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
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
Friday, January 31, 2020
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
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
Solving Quadratic Equation
Solving by Completing the Square
Friday, January 31, 2020
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
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
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
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.
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
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
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
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
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
Friday, January 31, 2020
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
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
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
Solving Quadratic Equation
Solve by quadratic formula
1.) x2 – 3x – 2 = 0
2.) 2x2 + 13 = 8x
3.) 3x2 – 3 = 4x
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
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
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