https://r2.erweima.ai/mathbot/image/1726985196675kuvqb0nx.jpg

Let's work through the factorization of the given algebraic expression:

\[
E = (x^2 - 2x)(m^2 - 49) - (49 - m^2)(7x + 6)
\]

### Step 1: Analyze the terms
Notice that both terms in the expression involve factors related to \( m^2 - 49 \), which can be factored using the difference of squares:
\[
m^2 - 49 = (m - 7)(m + 7)
\]
Thus, we can rewrite the expression as:
\[
E = (x^2 - 2x)(m - 7)(m + 7) - (49 - m^2)(7x + 6)
\]
Since \( 49 - m^2 \) is the same as \( -(m^2 - 49) \), we can rewrite the second part as:
\[
-(m - 7)(m + 7)(7x + 6)
\]

### Step 2: Factor common terms
Now the expression looks like:
\[
E = (m - 7)(m + 7) \left( (x^2 - 2x) - (7x + 6) \right)
\]

### Step 3: Simplify the inner expression
We need to simplify:
\[
(x^2 - 2x) - (7x + 6)
\]
Distribute the negative sign:
\[
x^2 - 2x - 7x - 6 = x^2 - 9x - 6
\]

### Step 4: Final factored form
Thus, the expression is factored as:
\[
E = (m - 7)(m + 7)(x^2 - 9x - 6)
\]
At this point, you could further attempt to factor the quadratic \( x^2 - 9x - 6 \), but it doesn't factor neatly with integer values. So the final factored form is:

\[
E = (m - 7)(m + 7)(x^2 - 9x - 6)
\]

Would you like more details or have any questions?

Here are five questions related to this problem:
1. What is the difference of squares, and how is it applied in this factorization?
2. How do you recognize when to factor out a common binomial expression?
3. What are some techniques for factoring quadratic expressions like \( x^2 - 9x - 6 \)?
4. How can the distributive property help simplify algebraic expressions?
5. What role does the negative sign play when distributing across binomials?

**Tip:** Always check for common factors in all terms before proceeding to factor binomials or quadratics.

Factoriza la siguiente expresión algebraica: E = (x^2 - 2x)(m^2 - 49) - (49 - m^2)(7x + 6)

Learn how to factor the algebraic expression E = (x^2 - 2x)(m^2 - 49) - (49 - m^2)(7x + 6) by applying the difference of squares and simplifying quadratic terms. This problem is suitable for high school students in Grades 9-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation