8.1 – solving quadratic equations
DESCRIPTION
8.1 – Solving Quadratic Equations. Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. Examples :. 5x 2 + 55 = 0. x 2 = 20. ( 3x – 1) 2 = – 4. ( x + 2) 2 = 18. x 2 + 8x = 1. 2x 2 – 2x + 7 = 0. 8.1 – Solving Quadratic Equations. - PowerPoint PPT PresentationTRANSCRIPT
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√‾ 2x = –2 ± 3√‾ 2
–4
( 3x – 1)2 = –4
3x – 1 = ±√‾‾3x – 1 = ± 2i
3x = 1 ± 2i
321 i
x
ix32
31
Review:
8.1 – Solving Quadratic EquationsCompleting the Square
( x + 3)2
x2 + 2(3x) + 9
x2 + 6x
26 23
x2 + 6x + 9
3 9
x2 + 6x + 9
( x + 3) ( x + 3)
( x + 3)2
x2 – 14x
214 277 49
x2 – 14x + 49
( x – 7) ( x – 7)
( x – 7)2
8.1 – Solving Quadratic EquationsCompleting the Square
x2 + 9x
29
2
29
481
x2 – 5x
481
92 xx
29
29
xx
2
29
x
25
2
25
425
425
52 xx
25
25
xx
2
25
x
8.1 – Solving Quadratic EquationsCompleting the Square
x2 + 8x = 1
28
24 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
22 21 1
55
53
1 x 55
52
1 2 x
53
1 2 x
53
1 x
53
1x
1
5x2 – 10x = –2
52
510
55 2
xx
52
22 xx
152
122 xx
53
1 2 x
515
1x
5155
x
or
8.1 – Solving Quadratic EquationsCompleting the Square
2x2 – 2x + 7 = 0
21
2
21
41
213
21 i
x 41
414
21 2
x
413
21 2
x
413
21
x
213
21
x
21
2x2 – 2x = –7
27
22
22 2
xx
272 xx
41
27
412 xx
413
21 2
x
2131 i
x
or
The quadratic formula is used to solve any quadratic equation.
2 42
x cb b aa
The quadratic formula is:
Standard form of a quadratic equation is: 2 0x xba c
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 42
x cb b aa
8.2 – Solving Quadratic EquationsThe Quadratic Formula
02 cbxax
cbxax 2
ac
xab
xaa
2
ac
xab
x
2
ab
ab
221
2
22
42 ab
ab
ac
ab
ab
xab
x 2
2
2
22
44
aa
ac
ab
ab
xab
x44
44 2
2
2
22
2 42
x cb b aa
8.2 – Solving Quadratic EquationsThe Quadratic Formula
22
2
2
22
44
44 aac
ab
ab
xab
x
2
2
2
22
44
4 aacb
ab
xab
x
2
2
2
22
44
4 aacb
ab
xab
x
2
22
44
2 aacb
ab
x
2
2
4
42 a
acbab
x
aacb
ab
x2
42
2
aacb
ab
x2
42
2
aacbb
x2
42
The quadratic formula is used to solve any quadratic equation.
2 42
x cb b aa
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 42
x cb b aa
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 42
x cb b aa
2 0x xba c
2 3 2 0x x a 1 c b 3 2
23 3 1 2412
x
3 9 82
x
3 12
x
3 12
x
3 12
x
3 12
x
42
x
2x
22
x
1x 3 12
x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 42
x cb b aa
2 0x xba c
22 5 0x x
a 2 c b 1 5
2 422
1 521x
1 1 404
x
1 414
x
8.2 – Solving Quadratic EquationsThe Quadratic Formula
2 42
x cb b aa
8.2 – Solving Quadratic EquationsThe Quadratic Formula
44 2 xx
044 2 xx
42
44411 2 x
86411
x
8631
x
8631 i
x
8391
i
x
8731 i
x
ix8
7381
2 42
x cb b aa
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
ZeroNegative
Two real solutionsOne real solutions
Two complex, non-real solutions
2 42
x cb b aa
8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate
2143 2
89
b2 – 4ac Name and Type of SolutionPositive
ZeroNegative
Two real solutionsOne 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 42
x cb b aa
8.2 – Solving Quadratic EquationsThe Quadratic Formula and the Discriminate
4441 2
641
b2 – 4ac Name and Type of SolutionPositive
ZeroNegative
Two real solutionsOne real solutions
Two complex, non-real solutions
a c b
63
Negative
Two complex, non-real solutions
044 2 xx
4 1 4
ix8
7381
2 42
x cb b aa
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 42
x cb b aa
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 = 02(x2 + 2x – 198) = 0
2 42
x cb b aa
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
279242
x
27962
x
2 42
x cb b aa
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
22.282
x2
2.282 x
22.26
x
1.13x
22.30
x
1.15xft
2 42
x cb b aa
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