Many quadratic equations can not be solved by factoring. Other techniques are required to solve them. 8.1 – Solving Quadratic Equations x 2 = 20 5x 2 + 55 = 0 Example s: ( x + 2) 2 = 18 ( 3x – 1) 2 = –4 x 2 + 8x = 1 2x 2 – 2x + 7 = 0 2 2 5 0 x x 4 4 2 x x
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
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 = ±√¯‾.
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.