x^2+2z-15 over 5z^2-20z-60 multiplied by z^2+8z+12 over z^2-9


We are tasked with simplifying the following expression:

\[
\frac{x^2 + 2z - 15}{5z^2 - 20z - 60} \times \frac{z^2 + 8z + 12}{z^2 - 9}
\]

### Step 1: Factor each expression.

1. **Factor \( 5z^2 - 20z - 60 \):**

\[
5z^2 - 20z - 60 = 5(z^2 - 4z - 12)
\]

Now factor \( z^2 - 4z - 12 \) by finding two numbers that multiply to \(-12\) and add up to \(-4\). However, \( z^2 - 4z - 12 \) cannot be factored easily over the integers, so we leave it as it is.

Thus, we have:
\[
5z^2 - 20z - 60 = 5(z^2 - 4z - 12)
\]

2. **Factor \( z^2 + 8z + 12 \):**

We need two numbers that multiply to \(12\) and add up to \(8\). These numbers are \(6\) and \(2\), so:

\[
z^2 + 8z + 12 = (z + 6)(z + 2)
\]

3. **Factor \( z^2 - 9 \):**

This is a difference of squares:
\[
z^2 - 9 = (z - 3)(z + 3)
\]

### Step 2: Rewrite the expression with the factored terms.

\[
\frac{x^2 + 2z - 15}{5(z^2 - 4z - 12)} \times \frac{(z + 6)(z + 2)}{(z - 3)(z + 3)}
\]

### Step 3: Simplify the expression.

Since none of the factors cancel out directly, the final simplified expression remains as:

\[
\frac{(x^2 + 2z - 15)(z + 6)(z + 2)}{5(z^2 - 4z - 12)(z - 3)(z + 3)}
\]

This is the most simplified form of the given expression.

---

Would you like more details on any step, or have further questions?

### Related Questions:
1. How do you factor a quadratic trinomial?
2. What is the difference of squares formula?
3. How do you simplify rational expressions involving multiplication?
4. What are common factoring techniques for quadratic expressions?
5. How do you identify when a quadratic expression is not factorable over integers?

### Tip:
Always look for common factors and use factoring techniques like grouping, the difference of squares, and the quadratic formula to simplify complex algebraic expressions.

x^2+2z-15 over 5z^2-20z-60 multiplied by z^2+8z+12 over z^2-9

Learn how to simplify the rational expression (x^2 + 2z - 15) / (5z^2 - 20z - 60) multiplied by (z^2 + 8z + 12) / (z^2 - 9) using factoring techniques such as difference of squares and factoring quadratic trinomials. This step-by-step solution is suitable 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