Equivalent expressions have the same value for every input in their shared domain. SAT questions reward strategic rewriting: factoring can expose zeros, completing the square can expose a vertex, and exponent rules can expose a growth factor.
Essential concepts
\[a^2-b^2=(a-b)(a+b) qquad a^m a^n=a^{m+n}\]
Recognize difference of squares and exponent structure before performing longer algebra.
SAT strategy and application
Factor to expose zeros
Rewrite \(4x^2-25\) as a product.
- Recognize squares
\(4x^2=(2x)^2\) and \(25=5^2\).
- Apply the identity
\(a^2-b^2=(a-b)(a+b)\).
Equivalent product: \((2x-5)(2x+5)\).
Mistakes and traps
Check your understanding
Which expression equals \(x^2+6x+9\)?
- \((x+3)^2\)
- \((x-3)^2\)
- \((x+9)(x+1)\)
- \(x(x+6)+3\)
Show answer and explanation
Answer: \((x+3)^2\)
\((x+3)^2=x^2+6x+9\) by expansion.
Key takeaways
What to remember
- Choose a form that reveals the requested feature.
- Use identities before beginning lengthy manipulation.
- Cancel common factors only, never isolated additive terms.
- Preserve original domain restrictions when simplifying rational expressions.
Continue learning
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.