For all real values of x such that x \neq 3, which of the following expressions is equival… — ACT

For all real values of $x$ such that $x \neq 3$ , which of the following expressions is equivalent to $\frac{x}{x - 3} - \frac{9}{(x - 3)^2}$ ?

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

To subtract the rational expressions, we must first find a common denominator. The denominators are

x-3

and

(x-3)^2

, so the least common denominator is

(x-3)^2

. We multiply the numerator and denominator of the first term by

x-3

, resulting in

\frac{x(x-3)}{(x-3)^2} - \frac{9}{(x-3)^2}

. Expanding the first numerator gives

x^2 - 3x

. Subtracting the second numerator gives

x^2 - 3x - 9

in the numerator. Expanding the denominator

(x-3)^2

yields

x^2 - 6x + 9

. Therefore, the simplified equivalent expression is

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

Identify the least common denominator (LCD) for the two rational expressions.

The LCD of

x - 3

and

(x - 3)^2

(x - 3)^2

To perform subtraction between two rational expressions, they must have a common denominator.

Rewrite the first expression with the common denominator by multiplying its numerator and denominator by

x - 3

\frac{x(x - 3)}{(x - 3)^2} = \frac{x^2 - 3x}{(x - 3)^2}

Multiplying the numerator and denominator by the same non-zero quantity preserves the value of the expression.

Subtract the numerators of the two expressions while keeping the common denominator.

\frac{x^2 - 3x - 9}{(x - 3)^2}

Once denominators are identical, subtract the numerators directly over the common denominator.

Expand the binomial in the denominator

(x - 3)^2

to match the standard polynomial form of the options.

(x - 3)^2 = x^2 - 6x + 9

, yielding

\frac{x^2 - 3x - 9}{x^2 - 6x + 9}

Expanding the denominator allows direct comparison with the polynomial choices provided.

Simplifying rational expressions by finding a common denominator

Question