https://file.aiquickdraw.com/mathbot/file/1727182171829liznbnuu.jpg

The given expression is:

\[
4x^2 - 9y^2 - 20x + 25
\]

We need to rewrite this expression in a factored form, as indicated in the equation.

### Step 1: Completing the square for \( x \)

We focus on the terms involving \( x \):

\[
4x^2 - 20x
\]

Factor out the \( 4 \):

\[
4(x^2 - 5x)
\]

Now, complete the square inside the parentheses. To complete the square for \( x^2 - 5x \), take half of \( -5 \), square it, and add and subtract this value inside the parentheses:

\[
x^2 - 5x + \left(\frac{-5}{2}\right)^2 - \left(\frac{-5}{2}\right)^2 = x^2 - 5x + \frac{25}{4} - \frac{25}{4}
\]

This simplifies to:

\[
4\left( (x - \frac{5}{2})^2 - \frac{25}{4} \right)
\]

Multiplying through by 4:

\[
4(x - \frac{5}{2})^2 - 25
\]

### Step 2: Combine with the \( y \)-terms

Now look at the \( y \)-terms in the expression:

\[
-9y^2 + 25
\]

This can be factored as a difference of squares:

\[
-9(y^2 - \frac{25}{9})
\]

This simplifies to:

\[
-9(y - \frac{5}{3})(y + \frac{5}{3})
\]

### Step 3: Final Factored Form

So, the expression can be rewritten as:

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

However, since we are using integers, simplify the coefficients:

\[
(2(x - 5) + 3y)(2(x - 5) - 3y)
\]

Now, filling in the boxes:

\[
(2(x - 5) + 3y)(2(x - 5) - 3y)
\]

So, the numbers to fill the boxes are:
- First box: 2
- Second box: 3
- Third box: 2
- Fourth box: 3

Would you like a detailed explanation of any part of this process?

Here are 5 related questions to explore:

1. How do you complete the square for a quadratic expression?
2. What are the steps for factoring a difference of squares?
3. How do you recognize perfect square trinomials in algebraic expressions?
4. How do coefficients affect the process of completing the square?
5. What are some other applications of completing the square in geometry?

**Tip:** Completing the square is a crucial technique in solving quadratic equations and simplifying expressions involving quadratics.

The expression 4x^2 - 9y^2 - 20x + 25 can be rewritten as [(2(x - 5) + 3y)][(2(x - 5) - 3y)]

Learn how to factor the expression 4x^2 - 9y^2 - 20x + 25 by completing the square and using the difference of squares. This step-by-step guide is perfect for students in Grades 9-11 looking to enhance their algebra skills.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation