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Solving Equations
To solve equations, there are various methods and techniques depending on the type of equation. Here are a few examples:
1. Solving Linear Equations: Linear equations are equations of the form `ax + b = c`, where `a`, `b`, and `c` are constants and `x` is the variable. To solve linear equations, you can use techniques like isolating the variable or using the properties of equality. For example, to solve the equation `3.5x + x = 180`, you can combine like terms and isolate the variable `x`:
``` 3.5x + x = 180 4.5x = 180 x = 180 / 4.5 x = 40 ```
Therefore, the solution to the equation `3.5x + x = 180` is `x = 40`.
2. Solving Quadratic Equations: Quadratic equations are equations of the form `ax^2 + bx + c = 0`, where `a`, `b`, and `c` are constants and `x` is the variable. There are different methods to solve quadratic equations, such as factoring, completing the square, or using the quadratic formula. For example, to solve the equation `x^2 - 4x - 5 = 0`, you can use the quadratic formula:
``` x = (-b ± √(b^2 - 4ac)) / (2a) ```
Plugging in the values `a = 1`, `b = -4`, and `c = -5`, we get:
``` x = (-(-4) ± √((-4)^2 - 4(1)(-5))) / (2(1)) x = (4 ± √(16 + 20)) / 2 x = (4 ± √36) / 2 x = (4 ± 6) / 2 ```
This gives us two solutions: `x = (4 + 6) / 2 = 5` and `x = (4 - 6) / 2 = -1`.
3. Solving Exponential Equations: Exponential equations involve variables in the exponent. To solve exponential equations, you can use logarithms. For example, to solve the equation `log_5(x + 121) = 4`, you can rewrite it as:
``` x + 121 = 5^4 x + 121 = 625 x = 625 - 121 x = 504 ```
Therefore, the solution to the equation `log_5(x + 121) = 4` is `x = 504`.
These are just a few examples of solving equations. There are many other types of equations and methods to solve them. If you have a specific equation you would like help with, please provide the equation, and I'll be happy to assist you further.
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