Algebraic Identities and Proofs

Understanding Algebraic Identities

• Definition: An algebraic identity is an equation that holds true for any value of the variable within it.
• Identity Symbol: Typically denoted with a symbol having three lines (≡) instead of the usual equal sign (=).
• Key Properties:
  • Cannot move terms across the identity symbol.
  • The left side must be manipulated to look like the right side to prove the identity.

Proving Algebraic Identities

• Objective: Make the left-hand side (LHS) of the equation identical to the right-hand side (RHS).
• Steps to Simplify and Prove:
  1. Simplify LHS:
     • Remove brackets.
     • Combine like terms.
  2. Rearrange LHS:
     • Make it look like the RHS.
     • Factorize if necessary.

Example Proving Process

1. Example 1:
   • Equation: Simplify LHS: (3(x-2) + 3x + 18)
   • Steps:
     • Expand brackets: (3x - 6 + 3x + 18)
     • Combine like terms: (6x + 12)
     • Factorize: (6(x+2)) to match RHS.
2. Example 2:
   • Equation: Simplify ((n+4)^2 - (n^2 + 2n))
   • Steps:
     • Expand ((n+4)^2): (n^2 + 4n + 4n + 16)
     • Expand other terms: (-n^2 - 2n)
     • Simplify: cancel (n^2), result (6n + 16)
     • Factorize: (2(3n+8)) to match RHS.

Additional Notes

• Alternative Scenarios:
  • Some proof questions might use the equal sign but require the same proving process as identity equations.

Conclusion

• Remember to simplify, factorize, and rearrange as necessary to prove identities.
• Always aim to make LHS exactly match RHS for proof completion.

Thank you for engaging with this topic. Happy studying!