Solving algebraic identities involves demonstrating that an equation holds true for all possible values of the variables involved. Here's a breakdown of the common methods:
1. Substitution:
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The most direct approach is to substitute various values for the variables. If both sides of the equation (Left-Hand Side or LHS, and Right-Hand Side or RHS) consistently yield the same result, the identity is likely valid.
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Example: Consider the identity (a + b)² = a² + 2ab + b². Let's test with a = 2 and b = 3.
- LHS: (2 + 3)² = 5² = 25
- RHS: 2² + 2(2)(3) + 3² = 4 + 12 + 9 = 25
- Since LHS = RHS, the identity holds true for these values. This should be tested with multiple sets of values to increase confidence.
2. Algebraic Manipulation:
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This involves transforming one side of the equation into the other using algebraic operations. The goal is to simplify one side until it matches the other.
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Common techniques include:
- Expanding brackets: Using the distributive property.
- Factoring: Expressing a term as a product of factors.
- Combining like terms: Simplifying expressions by adding or subtracting terms with the same variable and exponent.
- Adding/subtracting/multiplying/dividing both sides by the same value: Maintaining the equation's balance.
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Example: Prove the identity (a + b)² = a² + 2ab + b².
- Starting with the LHS: (a + b)²
- Expanding: (a + b)(a + b) = a(a + b) + b(a + b)
- Further expansion: a² + ab + ba + b²
- Since ab = ba (commutative property), we have a² + 2ab + b²
- This is equal to the RHS, thus proving the identity.
3. Using Known Identities:
- Employ established algebraic identities to simplify and transform expressions.
- Examples of common identities:
- (a + b)² = a² + 2ab + b²
- (a - b)² = a² - 2ab + b²
- (a + b)(a - b) = a² - b²
- (a + b)³ = a³ + 3a²b + 3ab² + b³
- (a - b)³ = a³ - 3a²b + 3ab² - b³
- a³ + b³ = (a + b)(a² - ab + b²)
- a³ - b³ = (a - b)(a² + ab + b²)
4. Combining Methods:
- Often, solving identities requires a combination of the above methods. You might use algebraic manipulation to simplify an expression and then substitute values to verify your result.
In summary, solving algebraic identities involves strategically manipulating expressions and substituting values to demonstrate that the equation holds true for all possible values of the variables.
