determine the $x$-intercept(s) of the following function. enter your answers as points in the form $(a, b)$. separate multiple $x$-intercepts with a comma. if there is no $x$-intercept, enter none.

\f(x) = \frac{x^2 + 12x + 35}{x^2 + 8x + 12}\

Question

determine the $x$-intercept(s) of the following function. enter your answers as points in the form $(a, b)$. separate multiple $x$-intercepts with a comma. if there is no $x$-intercept, enter none.

\f(x) = \frac{x^2 + 12x + 35}{x^2 + 8x + 12}\

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "blank_filling_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Rational Function Intercepts", "x-intercept" ], "new_concepts": [], "current_concepts": [ "Rational Function Intercepts", "x-intercept", "Domain of Rational Functions" ] } ## Set the numerator to zero and solve for x $$ \begin{aligned} x^2 + 12x + 35 &= 0 \ (x + 7)(x + 5) &= 0 \ x = -7 \quad &\text{or} \quad x = -5 \end{aligned} $$ ## Check the domain restrictions from the denominator $$ \begin{aligned} x^2 + 8x + 12 & e 0 \ (x + 6)(x + 2) & e 0 \ x e -6 \quad &\text{and} \quad x e -2 \end{aligned} $$ ## Determine the x-intercepts as coordinate points $$ \begin{aligned} x = -7 \implies (-7, 0) \ x = -5 \implies (-5, 0) \end{aligned} $$ Determine the $x$-intercept(s) of the following function. Enter your answers as points in the form $(a, b)$. Separate multiple $x$-intercepts with a comma. If there is no $x$-intercept, enter None. $$f(x) = \frac{x^2 + 12x + 35}{x^2 + 8x + 12}$$ The $x$-intercept(s) are $(-7, 0), (-5, 0)$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Algebra", "Rational Function Intercepts" ] }

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QUESTION IMAGE

Question

Explanation:

Set the numerator to zero and solve for x

Check the domain restrictions from the denominator

Determine the x-intercepts as coordinate points

Answer: