https://file.aiquickdraw.com/mathbot/file/172601985454652wqgf5e.jpg

Let's break down and solve some of the problems from the image you shared.

### Question 1: Find the square of the following expression
\[ a(x^2 + 2) \]

- The square of a binomial expression \((a + b)^2\) is given by:
  \[
  (a + b)^2 = a^2 + 2ab + b^2
  \]
  Here, \(x^2 + 2\) is a binomial expression. So,
  \[
  \left( x^2 + 2 \right)^2 = x^4 + 2(2)(x^2) + 4 = x^4 + 4x^2 + 4
  \]
  Multiply by \(a^2\):
  \[
  a^2(x^4 + 4x^2 + 4)
  \]

### Question 2: Write the following expression in whole square form
\[ 4x^2 - 20xy + 25y^2 \]

This is a quadratic expression and can be factored as a perfect square trinomial:
\[
4x^2 - 20xy + 25y^2 = (2x - 5y)^2
\]

### Question 3: Simplify
\[ (3x - 2y)^2 + (3y - 2x)^2 \]

Using the identity \((a - b)^2 = a^2 - 2ab + b^2\):
\[
(3x - 2y)^2 = 9x^2 - 12xy + 4y^2
\]
\[
(3y - 2x)^2 = 9y^2 - 12xy + 4x^2
\]
Now sum the two expressions:
\[
(3x - 2y)^2 + (3y - 2x)^2 = (9x^2 + 4x^2) + (9y^2 + 4y^2) - 12xy - 12xy
\]
\[
= 13x^2 + 13y^2 - 24xy
\]

Would you like details for any specific problem or further clarifications?

### Five Related Questions:
1. How do you identify a perfect square trinomial in any given expression?
2. Can you explain the process of expanding binomials for other forms like \((a + b)^3\)?
3. How does one simplify complex quadratic expressions?
4. What is the general rule for factoring trinomials like \(ax^2 + bx + c\)?
5. How can we use identities like \((a - b)^2\) in simplifying expressions efficiently?

### Tip:
Always look for patterns in algebraic expressions, like perfect squares or common terms, to simplify the process of factoring or expanding!

Find the square of the following expression, write the following expression in whole square form, and simplify expressions such as (3x - 2y)^2 + (3y - 2x)^2.

Learn how to find the square of algebraic expressions, factor trinomials into whole square form, and simplify quadratic expressions using binomial expansion. This tutorial provides step-by-step solutions to questions like (3x - 2y)^2 + (3y - 2x)^2, suitable for students in Grade 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation