Math Problem Statement

Solution

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Here are some common math problems and their solutions:

1. Simplifying Algebraic Expressions

Example: Simplify 3x2+2x5+4x23x+73x^2 + 2x - 5 + 4x^2 - 3x + 7.

Solution: Combine like terms: 3x2+4x2+2x3x5+7=7x2x+2.3x^2 + 4x^2 + 2x - 3x - 5 + 7 = 7x^2 - x + 2.

2. Solving Linear Equations

Example: Solve 2x+3=112x + 3 = 11.

Solution: Subtract 3 from both sides: 2x=8.2x = 8. Divide by 2: x=4.x = 4.

3. Solving Quadratic Equations

Example: Solve x25x+6=0x^2 - 5x + 6 = 0.

Solution: Factor the quadratic equation: (x2)(x3)=0.(x - 2)(x - 3) = 0. Set each factor to zero: x2=0x=2.x - 2 = 0 \Rightarrow x = 2. x3=0x=3.x - 3 = 0 \Rightarrow x = 3.

4. Simplifying Rational Expressions

Example: Simplify x24x22x\frac{x^2 - 4}{x^2 - 2x}.

Solution: Factor both numerator and denominator: (x2)(x+2)x(x2).\frac{(x - 2)(x + 2)}{x(x - 2)}. Cancel the common factor: x+2x.\frac{x + 2}{x}.

5. Solving Systems of Equations

Example: Solve the system {2x+y=5xy=1\begin{cases} 2x + y = 5 \\ x - y = 1 \end{cases}.

Solution: Add the equations: 3x=6x=2.3x = 6 \Rightarrow x = 2. Substitute xx into the second equation: 2y=1y=1.2 - y = 1 \Rightarrow y = 1.

6. Exponent Rules

Example: Simplify (2x3y2)2(2x^3 y^2)^2.

Solution: Apply the exponent rule: 22(x3)2(y2)2=4x6y4.2^2 (x^3)^2 (y^2)^2 = 4x^6 y^4.

7. Logarithmic Equations

Example: Solve log(x+1)=2\log(x + 1) = 2.

Solution: Rewrite the logarithmic equation in exponential form: x+1=102.x + 1 = 10^2. x+1=100x=99.x + 1 = 100 \Rightarrow x = 99.

8. Trigonometric Identities

Example: Simplify sin2(x)+cos2(x)\sin^2(x) + \cos^2(x).

Solution: Use the Pythagorean identity: sin2(x)+cos2(x)=1.\sin^2(x) + \cos^2(x) = 1.


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Math Problem Analysis

Mathematical Concepts

Algebra
Linear Equations
Quadratic Equations
Rational Expressions
Systems of Equations
Exponent Rules
Logarithmic Equations
Trigonometric Identities

Formulas

Quadratic formula
Exponent rules
Logarithmic equations
Pythagorean identity

Theorems

Pythagorean theorem

Suitable Grade Level

High School