many quadratic equations can not be solved by factoring. other techniques are required to solve...
TRANSCRIPT
Many quadratic equations can not be solved by factoring. Other techniques are required to solve them.
8.1 – Solving Quadratic Equations
x2 = 20 5x2 + 55 = 0
Examples:
( x + 2)2 = 18 ( 3x – 1)2 = –4
x2 + 8x = 1 2x2 – 2x + 7 = 0
22 5 0x x 44 2 xx
If b is a real number and if a2 = b, then a = ±√¯‾.
20
8.1 – Solving Quadratic EquationsSquare Root Property
b
x2 = 20
x = ±√‾‾
x = ±√‾‾‾‾4·5
x = ± 2√‾ 5 –11
5x2 + 55 = 0
x = ±√‾‾‾
5x2 = –55
x2 = –11
x = ± i√‾‾‾11
If b is a real number and if a2 = b, then a = ±√¯‾.
18
8.1 – Solving Quadratic EquationsSquare Root Property
b
( x + 2)2 = 18
x + 2 = ±√‾‾
x + 2 = ±√‾‾‾‾9·2
x +2 = ± 3√‾ 2
x = –2 ± 3√‾ 2
–4
( 3x – 1)2 = –4
3x – 1 = ±√‾‾
3x – 1 = ± 2i
3x = 1 ± 2i
3
21 ix
ix3
2
3
1
Review:
8.1 – Solving Quadratic EquationsCompleting the Square
( x + 3)2
x2 + 2(3x) + 9
x2 + 6x
2
6 23
x2 + 6x + 9
3 9
x2 + 6x + 9
( x + 3) ( x + 3)
( x + 3)2
x2 – 14x
2
14 277 49
x2 – 14x + 49
( x – 7) ( x – 7)
( x – 7)2
8.1 – Solving Quadratic EquationsCompleting the Square
x2 + 9x
2
9
2
2
94
81
x2 – 5x
4
8192 xx
2
9
2
9xx
2
2
9
x
2
5
2
2
54
25
4
2552 xx
2
5
2
5xx
2
2
5
x
8.1 – Solving Quadratic EquationsCompleting the Square
x2 + 8x = 1
2
824 16
1611682 xx
174 2 x
174 2 x
174 x
174 x
4
x2 + 8x = 1
8.1 – Solving Quadratic EquationsCompleting the Square
5x2 – 10x + 2 = 0
2
2 21 1
5
5
5
31 x
5
5
5
21 2 x
5
31 2 x
5
31 x
5
31x
1
5x2 – 10x = –2
5
2
5
10
5
5 2
xx
5
222 xx
15
2122 xx
5
31 2 x
5
151x
5
155x
or
8.1 – Solving Quadratic EquationsCompleting the Square
2x2 – 2x + 7 = 0
2
1
2
2
1
4
1
2
13
2
1 ix 4
1
4
14
2
12
x
4
13
2
12
x
4
13
2
1 x
2
13
2
1 x
2
1
2x2 – 2x = –7
2
7
2
2
2
2 2
xx
2
72 xx
4
1
2
7
4
12 xx
4
13
2
12
x
2
131 ix
or
The quadratic formula is used to solve any quadratic equation.
2 4
2x
cb b a
a
The quadratic formula is:
Standard form of a quadratic equation is:2 0x xba c
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula
02 cbxax
cbxax 2
a
cx
a
bx
a
a 2
a
cx
a
bx
2
a
b
a
b
22
1 2
22
42 a
b
a
b
a
c
a
b
a
bx
a
bx
2
2
2
22
44
a
a
a
c
a
b
a
bx
a
bx
4
4
44 2
2
2
22
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula
22
2
2
22
4
4
44 a
ac
a
b
a
bx
a
bx
2
2
2
22
4
4
4 a
acb
a
bx
a
bx
2
2
2
22
4
4
4 a
acb
a
bx
a
bx
2
22
4
4
2 a
acb
a
bx
2
2
4
4
2 a
acb
a
bx
a
acb
a
bx
2
4
2
2
a
acb
a
bx
2
4
2
2
a
acbbx
2
42
The quadratic formula is used to solve any quadratic equation.
2 4
2x
cb b a
a
The quadratic formula is:
Standard form of a quadratic equation is: 2 0x xba c
2 4 8 0x x
a 1 c b4 8
23 5 6 0x x
a 3 c b 5
22 0x x
a 2 c b1 0
2 10x a 1 c b0 106
2 10 0x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 4
2x
cb b a
a
2 0x xba c
2 3 2 0x x
2x 1x
1x 2x 0
1 0x 2 0x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 4
2x
cb b a
a
2 0x xba c
2 3 2 0x x a 1 c b 3 2
23 3 1 24
12x
3 9 8
2x
3 1
2x
3 1
2x
3 1
2x
3 1
2x
4
2x
2x
2
2x
1x 3 1
2x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 4
2x
cb b a
a
2 0x xba c
22 5 0x x
a 2 c b 1 5
24
22
1 521x
1 1 40
4x
1 41
4x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula
44 2 xx
044 2 xx
42
44411 2 x
8
6411 x
8
631 x
8
631 ix
8
391
ix
8
731 ix
ix
8
73
8
1
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate
The discriminate is the radicand portion of the quadratic formula (b2 – 4ac).
It is used to discriminate among the possible number and type of solutions a quadratic equation will have.
b2 – 4ac Name and Type of SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate
2143 2
89
b2 – 4ac Name and Type of SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
2 3 2 0x x a 1 c b 3 2
1
Positive
Two real solutions
2x 1x
2 4
2x
cb b a
a
8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate
4441 2
641
b2 – 4ac Name and Type of SolutionPositive
Zero
Negative
Two real solutions
One real solutions
Two complex, non-real solutions
a c b
63
Negative
Two complex, non-real solutions
044 2 xx
4 1 4
ix8
73
8
1
2 4
2x
cb b a
a
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
2 4
2x
cb b a
a
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
(x + 2)2 + x2 = 202
x2 + 4x + 4 + x2 = 400
2x2 + 4x + 4 = 400
2x2 + 4x – 369 = 0
2(x2 + 2x – 198) = 0
2 4
2x
cb b a
a
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
2(x2 + 2x – 198) = 0
12
1981422 2 x
2
79242 x
2
7962 x
2 4
2x
cb b a
a
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
2
7962x
2
2.282
2
2.282 x
2
2.282 x
2
2.26x
1.13x
2
2.30x
1.15xft
2 4
2x
cb b a
a
8.2 – Solving Quadratic Equations
The Quadratic Formula
Given the diagram below, approximate to the nearest foot how many feet of walking distance a person saves by cutting across the lawn instead of walking on the sidewalk.
20 feet
x + 2
x
The Pythagorean Theorem
a2 + b2 = c2
1.13x
ft2.28
ft
21.131.13
28 – 20 = 8 ft