Analyzing the rational expression 38 over 25 minus x squared

Key takeaway: This expression is a rational function with vertical asymptotes at the zeros of the denominator.

What the expression is

The expression

$\frac{38}{25-x^2}$

is a rational expression, meaning it is a quotient of two polynomials. The denominator factors as

$25-x^2=(5-x)(5+x).$

So the expression can also be written as

$\frac{38}{(5-x)(5+x)}.$

Domain and undefined values

The denominator cannot be zero. Solve

$25-x^2=0 \quad \Rightarrow \quad x^2=25 \quad \Rightarrow \quad x=\pm 5.$

Therefore the expression is undefined at $x=5$ and $x=-5$.

That means its domain is all real numbers except $\pm 5$.

Important algebraic feature

Because the numerator is a constant and the denominator is quadratic, this expression has vertical asymptotes at $x=\pm 5$. It does not simplify further unless it is part of a larger expression where factors cancel. Here, nothing cancels because the numerator is $38$, not a factor containing $(5-x)$ or $(5+x)$.

Why this matters

Many students look only for cancellation and forget the domain restriction. Even if a later step seems to produce a simpler form, you must still exclude the values that make the original denominator zero.

This expression is also useful in calculus or graphing contexts because its symmetry depends on $x^2$. Since the denominator contains $x^2$, the graph is even: replacing $x$ by $-x$ gives the same value.

Key points to remember

• Factor the denominator as $(5-x)(5+x)$.
• Exclude $x=\pm 5$ from the domain.
• No cancellation is possible with the constant numerator.
• The function is symmetric about the $y$-axis.

If you are preparing for a follow-up question, the next step is usually to evaluate, graph, or differentiate this rational function, not to simplify it further.

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Pitfalls the pros know 👇 A common mistake is to factor $25-x^2$ incorrectly as $(25-x)(25+x)$. The correct difference of squares factorization is $(5-x)(5+x)$. Another frequent error is forgetting that the expression is undefined at $x=\pm 5$. Even when the fraction does not simplify, those excluded values still matter because they belong to the original denominator. If this expression appears inside an equation or inequality, the domain restriction must be carried through every step.

What if the problem changes? If the expression changes to $\frac{38}{25-x^2}=2$, then you would solve a rational equation instead of just analyzing the fraction. Multiply both sides by $25-x^2$, but first note that $x\neq \pm 5$. The equation becomes $38=2(25-x^2)$, which simplifies to $2x^2=12$ and then $x=\pm\sqrt{6}$. The same domain restrictions still apply. This shows how the original rational expression becomes part of a larger algebraic problem.

Tags: difference of squares, domain restrictions, vertical asymptotes