Evaluating Expressions

To evaluate an expression, replace each variable with its value and compute.

The single habit that prevents most errors: put the substituted value in parentheses. For 3x² with x = −2, write 3(−2)². Without the parentheses you are likely to write 3 × −2² and get −12 instead of 12.

After substituting, it is just order of operations. Parentheses, exponents, multiply and divide, add and subtract.

Evaluation is applying the expression as a function to a specific input. The substitution must respect the expression tree, and parentheses are how you preserve that tree when the substituted value is not a bare positive number.

The exponent case is the one that matters: in 3x² the square applies to x alone, so at x = −2 the value is 3·(−2)² = 3·4 = 12. Dropping the parentheses changes the tree to −(3·2²) and the answer to −12. Similar care is needed with substituted fractions and with expressions substituted into other expressions, which becomes function composition later.

For multivariable expressions, substitute every occurrence of each variable simultaneously; substituting one variable's value into an expression that still contains it elsewhere is a common source of silent errors.