Question

Simplify the following expression completely.
start fraction, x, squared, plus, 3, x, minus, 28, divided by, x, squared, plus, 14, x, plus, 49, end fraction
x
2
+14x+49
x
2
+3x−28
​

Answer 1

To simplify the expression \(\frac{x^2 + 3x - 28}{x^2 + 14x + 49}\), we need to factor both the numerator and the denominator.

**Step 1: Factor the numerator \(x^2 + 3x - 28\)**

We are looking for two numbers that multiply to \(-28\) and add to \(3\). The numbers \(7\) and \(-4\) work:

\[
x^2 + 3x - 28 = (x + 7)(x - 4)
\]

**Step 2: Factor the denominator \(x^2 + 14x + 49\)**

We are looking for two numbers that multiply to \(49\) and add to \(14\). The numbers \(7\) and \(7\) work:

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

**Step 3: Simplify the expression**

Now substitute the factored forms into the original expression:

\[
\frac{(x + 7)(x - 4)}{(x + 7)(x + 7)}
\]

We can cancel the common factor \((x + 7)\) from the numerator and the denominator, provided that \(x + 7 \neq 0\) (i.e., \(x \neq -7\)):

\[
\frac{x - 4}{x + 7}
\]

**Final Result:**

Thus, the simplified expression is:

\[
\frac{x - 4}{x + 7}, \quad x \neq -7
\]