Simplify the expression completely if possible.
start fraction, x, squared, plus, 14, x, plus, 49, divided by, x, squared, plus, 16, x, plus, 63, end fraction
x
2
+16x+63
x
2
+14x+49
​

Let's simplify the expression:

\[
\frac{x^2 + 14x + 49}{x^2 + 16x + 63}
\]

### Step 1: Factor the numerator and the denominator.
- **Numerator:** \( x^2 + 14x + 49 \)

This is a perfect square trinomial, and it can be factored as:

\[
x^2 + 14x + 49 = (x + 7)^2
\]

- **Denominator:** \( x^2 + 16x + 63 \)

We need to find two numbers that multiply to 63 and add to 16. These numbers are 9 and 7, so the denominator factors as:

\[
x^2 + 16x + 63 = (x + 7)(x + 9)
\]

### Step 2: Simplify the expression.
Now substitute the factored forms back into the original expression:

\[
\frac{(x + 7)^2}{(x + 7)(x + 9)}
\]

Cancel out one \( (x + 7) \) from the numerator and the denominator:

\[
\frac{x + 7}{x + 9}
\]

### Final Answer:
The simplified expression is:

\[
\frac{x + 7}{x + 9}
\]

Would you like more details or have any questions about this solution? Here are five related questions you might consider:

1. How do you factor quadratic expressions in general?
2. Can all quadratic expressions be factored easily?
3. What happens if \( x = -9 \) in this expression?
4. How do perfect square trinomials differ from other quadratics?
5. How would you graph the simplified expression?

**Tip:** When simplifying rational expressions, always check for common factors between the numerator and denominator to cancel them out!

Simplify the expression completely if possible: start fraction, x, squared, plus, 14, x, plus, 49, divided by, x, squared, plus, 16, x, plus, 63, end fraction.

Discover how to simplify the rational expression (x^2 + 14x + 49) / (x^2 + 16x + 63) using factoring techniques. This step-by-step guide explains perfect square trinomials and how to cancel common factors in rational expressions. Perfect for students in Grades 8-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation