Equivalent forms reveal different features. Choose the form that exposes the quantity requested, and preserve the original expression's restrictions.
An equivalent-expression question asks for the same mathematical value written another way. The useful transformation is the one that exposes the feature the prompt asks about, rather than expanding everything automatically.
Worked example: factoring reveals zeros
These are original teaching exercises, not released SAT questions. The expression x² − 5x + 6 equals (x − 2)(x − 3), since expanding gives x² − 3x − 2x + 6. The factored form exposes zeros at x = 2 and x = 3.
If the question asks for a coefficient in standard form, expansion may be useful instead. A form is useful because of what it reveals, not because it is always shorter.
Worked example: completing the square reveals a minimum
x² − 6x + 11 = (x − 3)² + 2. Expanding the right side gives x² − 6x + 9 + 2. Since a real square is nonnegative, the expression has minimum value 2, reached when x = 3.
The zero-product rule cannot be applied directly to a sum such as (x − 3)² + 2. Use the structure appropriate to the question.
Worked example: cancellation preserves a restriction
(x² − 9)/(x − 3) simplifies to x + 3 for x ≠ 3. The original is undefined at 3, although the simplified polynomial has a value there. Cancelling a factor does not add the excluded input to the original domain.
Only factors cancel: (x + 3)/x is not equal to 3, because x is not a common factor of the entire numerator.
Worked example: exponents and radicals
For x > 0, x^(3/2) = x√x. This follows from x^(3/2) = x^1 × x^(1/2). For any real x, √(x²) = |x|, so replacing it with x without a sign condition is unsafe. For example, x = −4 gives √16 = 4.
Also check exponent operations: x² × x³ = x⁵, while (x²)³ = x⁶. Addition of expressions does not follow the multiplication rule.
Verify the form rather than relying on a numerical coincidence
Expand or factor to establish the identity. Testing a convenient input can expose an error, but agreement at one input does not prove two expressions are equivalent for all permitted values.
Keep a practice log of sign, factor-versus-term, exponent, and domain errors. For solving equations after rewriting them, read nonlinear equations and systems.
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