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Here are some common math problems and their solutions:

### 1. Simplifying Algebraic Expressions
**Example: Simplify \(3x^2 + 2x - 5 + 4x^2 - 3x + 7\).**

**Solution:**
Combine like terms:
\[3x^2 + 4x^2 + 2x - 3x - 5 + 7 = 7x^2 - x + 2.\]

### 2. Solving Linear Equations
**Example: Solve \(2x + 3 = 11\).**

**Solution:**
Subtract 3 from both sides:
\[2x = 8.\]
Divide by 2:
\[x = 4.\]

### 3. Solving Quadratic Equations
**Example: Solve \(x^2 - 5x + 6 = 0\).**

**Solution:**
Factor the quadratic equation:
\((x - 2)(x - 3) = 0.\)
Set each factor to zero:
\[x - 2 = 0 \Rightarrow x = 2.\]
\[x - 3 = 0 \Rightarrow x = 3.\]

### 4. Simplifying Rational Expressions
**Example: Simplify \(\frac{x^2 - 4}{x^2 - 2x}\).**

**Solution:**
Factor both numerator and denominator:
\[\frac{(x - 2)(x + 2)}{x(x - 2)}.\]
Cancel the common factor:
\[\frac{x + 2}{x}.\]

### 5. Solving Systems of Equations
**Example: Solve the system \(\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}\).**

**Solution:**
Add the equations:
\[3x = 6 \Rightarrow x = 2.\]
Substitute \(x\) into the second equation:
\[2 - y = 1 \Rightarrow y = 1.\]

### 6. Exponent Rules
**Example: Simplify \((2x^3 y^2)^2\).**

**Solution:**
Apply the exponent rule:
\[2^2 (x^3)^2 (y^2)^2 = 4x^6 y^4.\]

### 7. Logarithmic Equations
**Example: Solve \(\log(x + 1) = 2\).**

**Solution:**
Rewrite the logarithmic equation in exponential form:
\[x + 1 = 10^2.\]
\[x + 1 = 100 \Rightarrow x = 99.\]

### 8. Trigonometric Identities
**Example: Simplify \(\sin^2(x) + \cos^2(x)\).**

**Solution:**
Use the Pythagorean identity:
\[\sin^2(x) + \cos^2(x) = 1.\]

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Explore comprehensive solutions to common math problems including algebraic expressions, linear equations, quadratic equations, rational expressions, systems of equations, exponent rules, logarithmic equations, and trigonometric identities. Learn step-by-step how to solve each problem type with detailed explanations.

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