what is x squared minus 2x minus 15 over 10 minus 2x simplified?

To simplify the expression:

\[
\frac{x^2 - 2x - 15}{10 - 2x}
\]

we follow these steps:

### Step 1: Factor the numerator
The numerator is \(x^2 - 2x - 15\). We factor this quadratic expression.

To factor it, we need two numbers that multiply to \(-15\) (the constant term) and add to \(-2\) (the coefficient of \(x\)).

These two numbers are \(-5\) and \(3\), so we can factor the numerator as:

\[
x^2 - 2x - 15 = (x - 5)(x + 3)
\]

### Step 2: Factor the denominator
The denominator is \(10 - 2x\). We can factor out a 2:

\[
10 - 2x = 2(5 - x)
\]

### Step 3: Simplify the expression
Now the expression looks like this:

\[
\frac{(x - 5)(x + 3)}{2(5 - x)}
\]

Notice that \(5 - x\) can be rewritten as \(-(x - 5)\). This allows us to simplify:

\[
\frac{(x - 5)(x + 3)}{2(-(x - 5))} = \frac{(x - 5)(x + 3)}{-2(x - 5)}
\]

Since \(x - 5\) appears in both the numerator and denominator, we can cancel it out, assuming \(x \neq 5\) (to avoid division by zero):

\[
\frac{(x + 3)}{-2}
\]

### Final simplified expression:
\[
\frac{x + 3}{-2} \quad \text{or} \quad -\frac{x + 3}{2}
\]

This is the simplified form of the given expression.

Would you like further details on any step?

Here are 5 related questions to explore:

1. How do you factor quadratic expressions?
2. What is the significance of factoring in simplifying rational expressions?
3. How do you cancel common terms in fractions?
4. Why is it important to check for values that make the denominator zero?
5. How do you simplify complex fractions with binomials?

**Tip**: Always check for common factors in both the numerator and denominator when simplifying fractions.

What is x squared minus 2x minus 15 over 10 minus 2x simplified?

Explore how to simplify the rational expression (x^2 - 2x - 15)/(10 - 2x) by factoring the quadratic numerator and canceling common terms in the fraction. This solution covers key concepts like factoring quadratics and simplifying fractions, ideal for students in Grades 9-11.

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