-
Notifications
You must be signed in to change notification settings - Fork 0
Expand file tree
/
Copy pathmath9_3.html
More file actions
190 lines (171 loc) · 6.51 KB
/
Copy pathmath9_3.html
File metadata and controls
190 lines (171 loc) · 6.51 KB
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
102
103
104
105
106
107
108
109
110
111
112
113
114
115
116
117
118
119
120
121
122
123
124
125
126
127
128
129
130
131
132
133
134
135
136
137
138
139
140
141
142
143
144
145
146
147
148
149
150
151
152
153
154
155
156
157
158
159
160
161
162
163
164
165
166
167
168
169
170
171
172
173
174
175
176
177
178
179
180
181
182
183
184
185
186
187
188
189
190
<!DOCTYPE html>
<html lang="en">
<head>
<meta charset="UTF-8">
<meta name="viewport" content="width=device-width, initial-scale=1.0">
<title>Solving Equations Deeply</title>
<style>
* {
margin: 0;
padding: 0;
box-sizing: border-box;
}
body {
font-family: 'Arial', sans-serif;
background-color: #f8f9fa;
color: #495057;
line-height: 1.6;
padding: 20px;
}
header {
background-color: #007bff;
color: #fff;
padding: 20px;
text-align: center;
border-radius: 8px;
}
header h1 {
font-size: 2.5rem;
}
.content {
background-color: #ffffff;
padding: 30px;
margin-top: 20px;
border-radius: 8px;
box-shadow: 0 4px 8px rgba(0, 0, 0, 0.1);
max-width: 1200px;
margin-left: auto;
margin-right: auto;
}
h2 {
font-size: 2rem;
color: #007bff;
margin-bottom: 15px;
}
h3 {
font-size: 1.6rem;
color: #0056b3;
margin-top: 15px;
}
p, ul {
margin-bottom: 15px;
font-size: 1rem;
}
ul {
list-style-type: square;
padding-left: 20px;
}
footer {
text-align: center;
padding: 10px;
margin-top: 30px;
background-color: #007bff;
color: #fff;
border-radius: 8px;
}
footer p {
font-size: 0.9rem;
}
</style>
</head>
<body>
<header>
<h1>Solving Equations Deeply</h1>
</header>
<div class="content">
<h2>1. Linear Equations</h2>
<p>A <strong>linear equation</strong> is an equation of the form:</p>
<p><strong>Ax + B = 0</strong></p>
<p>Where:</p>
<ul>
<li>A and B are constants</li>
<li>x is the variable</li>
</ul>
<h3>Example:</h3>
<p><strong>2x + 3 = 0</strong></p>
<p><strong>Solution Steps:</strong></p>
<ol>
<li>Subtract 3 from both sides: 2x = -3</li>
<li>Divide by 2: x = -3/2</li>
</ol>
<p><strong>General Formula for Solving:</strong></p>
<p>For an equation <strong>Ax + B = 0</strong>, the solution is:</p>
<p><strong>x = -B/A</strong></p>
<h2>2. Quadratic Equations</h2>
<p>A <strong>quadratic equation</strong> is an equation of the form:</p>
<p><strong>Ax² + Bx + C = 0</strong></p>
<h3>Example:</h3>
<p><strong>x² - 5x + 6 = 0</strong></p>
<p><strong>Solution Steps:</strong></p>
<ol>
<li>Factor the quadratic expression: (x - 2)(x - 3) = 0</li>
<li>Set each factor equal to zero: x - 2 = 0 or x - 3 = 0</li>
<li>Solve for x: x = 2 or x = 3</li>
</ol>
<h3>General Formula:</h3>
<p>For a quadratic equation <strong>Ax² + Bx + C = 0</strong>, use the <strong>quadratic formula</strong>:</p>
<p><strong>x = (-B ± √(B² - 4AC)) / 2A</strong></p>
<p>Where:</p>
<ul>
<li>B² - 4AC is the <strong>discriminant</strong></li>
</ul>
<p>If the discriminant (Δ = B² - 4AC) is positive, there are two real roots. If it is zero, there is exactly one real root. If it is negative, there are two complex roots.</p>
<h2>3. Cubic Equations</h2>
<p>A <strong>cubic equation</strong> is an equation of the form:</p>
<p><strong>Ax³ + Bx² + Cx + D = 0</strong></p>
<h3>Example:</h3>
<p><strong>x³ - 6x² + 11x - 6 = 0</strong></p>
<p>Factoring it:</p>
<p><strong>(x - 1)(x - 2)(x - 3) = 0</strong></p>
<p>The solutions are: x = 1, x = 2, x = 3</p>
<h2>4. Systems of Equations</h2>
<p>Systems of equations involve solving for multiple variables at the same time. These can be <strong>linear systems</strong> (multiple linear equations) or <strong>nonlinear systems</strong> (involving quadratics, exponentials, etc.).</p>
<h3>Example:</h3>
<p>Consider the system of equations:</p>
<p><strong>2x + 3y = 6</strong></p>
<p><strong>4x - y = 5</strong></p>
<p><strong>Solution Steps:</strong></p>
<ol>
<li>Solve equation (2) for y: y = 4x - 5</li>
<li>Substitute into equation (1): 2x + 3(4x - 5) = 6</li>
<li>Simplify: 2x + 12x - 15 = 6</li>
<li>14x = 21</li>
<li>x = 21 / 14 = 3/2</li>
<li>Substitute x = 3/2 into y = 4x - 5: y = 6 - 5 = 1</li>
</ol>
<p>The solution is: x = 3/2, y = 1</p>
<h2>5. Exponential and Logarithmic Equations</h2>
<p>These equations involve exponents or logarithms and can be solved using properties of exponents or logarithms.</p>
<h3>Exponential Equation Example:</h3>
<p><strong>3^x = 81</strong></p>
<p>Take the logarithm of both sides:</p>
<p><strong>log(3^x) = log(81)</strong></p>
<p>Using the property log(a^b) = b log(a):</p>
<p>x log(3) = log(81)</p>
<p>Since 81 = 3^4, log(81) = 4 log(3), so:</p>
<p>x log(3) = 4 log(3)</p>
<p>x = 4</p>
<h2>6. Absolute Value Equations</h2>
<p>An <strong>absolute value equation</strong> involves the absolute value function, such as:</p>
<p><strong>|x - 3| = 5</strong></p>
<p>This means the expression inside the absolute value can be either 5 or -5:</p>
<ul>
<li>x - 3 = 5 → x = 8</li>
<li>x - 3 = -5 → x = -2</li>
</ul>
<p>The solutions are: x = 8 or x = -2</p>
<h2>7. Higher-Degree Polynomials</h2>
<p>Equations with powers higher than two can be solved by factoring (if possible), numerical methods, or using specific formulas like <strong>Descartes' Rule of Signs</strong> or <strong>Rational Root Theorem</strong>.</p>
<h2>Summary of Key Formulas:</h2>
<ul>
<li>Linear Equation: x = -B/A</li>
<li>Quadratic Equation (Quadratic Formula): x = (-B ± √(B² - 4AC)) / 2A</li>
<li>Exponential Equation: a^x = b → x = log_a(b)</li>
<li>Logarithmic Equation: log_b(x) = y → x = b^y</li>
</ul>
</div>
<footer>
<p>© 2025 Yoseph Feyisa Wegi | All Rights Reserved.</p>
</footer>
</body>
</html>