Advanced Math / Equivalent Expressions
Equivalent Expressions
Equivalent expressions have the same value for every allowed value of the variable. Common strategies include expanding, factoring, combining like terms, applying exponent rules, completing the square, and simplifying rational expressions while respecting denominator restrictions.
Least you need to know
- Special products such as difference of squares and perfect-square trinomials can make rewriting faster.
- Rational expressions can be simplified by canceling common factors, not individual terms.
- Exponent rules depend on whether powers are being multiplied, divided, or raised to another power.
Key notation
- $a^2-b^2$ — A difference of squares, equal to $(a-b)(a+b)$
- $(x-h)^2+k$ — Vertex or completed-square form of a quadratic expression
Worked example
- Consider $x^2+12x+40$.
- Half of 12 is 6, and $6^2=36$.
- Since $(x+6)^2=x^2+12x+36$, the expression is $(x+6)^2+4$.
Common mistakes
- Do not cancel terms separated by addition or subtraction in rational expressions.
- When completing the square, compare the constant term carefully.
- A negative exponent means a reciprocal, not a negative value.
How to recognize it
- The problem asks which expression is equivalent.
- The expression may contain factored forms, expanded forms, powers, roots, or fractions with polynomial numerators and denominators.
- The problem may ask for a missing coefficient or constant in an equivalent form.
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