Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original
expression.

Ask by Whittaker Page. in the United States

Mar 29,2025

Question

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original
expression.

Ask by Whittaker Page. in the United States

Mar 29,2025

UpStudy AI · Accepted Answer

**Step 1. Factor the numerator and denominator.**

Write the numerator $$x^2 - 6x + 9$$ as a perfect square:
$$
x^2 - 6x + 9 = (x-3)^2.
$$

Factor the denominator $$4x - 12$$ by taking out the common factor:
$$
4x - 12 = 4(x-3).
$$

**Step 2. Simplify the expression by canceling common factors.**

The original expression is:
$$
\frac{x^{2}-6 x+9}{4 x-12} = \frac{(x-3)^2}{4(x-3)}.
$$

Cancel the common factor $$(x-3)$$ from the numerator and the denominator. This cancellation is valid provided that $$(x-3) \neq 0$$, i.e. $$x\neq3$$:
$$
\frac{(x-3)^2}{4(x-3)} = \frac{x-3}{4} \quad \text{with} \quad x \neq 3.
$$

**Step 3. Determine the values to be excluded from the domain.**

The denominator cannot be zero in the original expression. Since:
$$
4(x-3) = 0 \quad \Longrightarrow \quad x-3=0 \quad \Longrightarrow \quad x=3,
$$
the value $$x=3$$ must be excluded from the domain.

**Final Answer:**

The simplified expression is
$$
\frac{x-3}{4},
$$
with the domain restriction $$x \neq 3$$.
The simplified expression is $$ \frac{x-3}{4} $$, and $$ x $$ cannot be 3.

Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original
expression.

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Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original expression.

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Simplify the rational expression. Find all numbers that must be excluded from the domain of the simplified rational expression in order for it to be equivalent to the original
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