Problem 90 - Algebra A Combined Function

Question

Write each integer as a product of two integers such that one of the factors is a perfect square. For example, write 18 as \(9 \cdot 2,\) because 9 is a perfect square. $$ 45 $$

Step-by-Step Solution

Answer

45 can be written as \( 9 \cdot 5 \).

Step 1: Identify Perfect Squares

List the perfect squares less than or equal to the given integer 45. The perfect squares are 1, 4, 9, 16, 25, 36, and 49. Note that 49 is greater than 45, so we won't consider it.

Step 2: Find Factor Pair Including a Perfect Square

Look for pairs of factors of 45 such that one factor is one of the perfect squares identified in Step 1. We know that 9 is a perfect square. Check if 9 is a factor of 45: \( \frac{45}{9} = 5 \). Thus, 45 can be expressed as \( 9 \cdot 5 \), where 9 is the perfect square.

Key Concepts

Understanding Perfect Squares

Perfect squares are numbers that result from multiplying an integer by itself. This is an important concept in mathematics as it relates to simplifying expressions and solving equations.
Here are some key examples of perfect squares:

• 4, since it's the result of 2 multiplied by 2, i.e., \(2^2\).
• 9, because 3 multiplied by 3 results in 9, or \(3^2\).
• 16, a product of 4 times 4, represented as \(4^2\).

Recognizing these patterns helps in tasks like factorization as it simplifies the process.
Identifying perfect squares is a crucial step when working with problems involving integer factorization.

The Role of Integer Pairs

Integer pairs are sets of two integers that multiply together to give a particular number. Finding these pairs is key to solving factorization problems.
When looking for integer pairs that include a perfect square, the challenge is to ensure one of the numbers is a perfect square.
For example:

• For 18, the integer pairs could be (1,18), (2,9), (3,6), but we focus on (9,2) since 9 is a perfect square.
• Similarly, for 45, possible pairs include (1,45), (3,15), (5,9), and we prioritize (9,5).

Working with integer pairs in this manner aids in recognizing relationships within numbers and solving various algebraic equations.

Effective Factorization Techniques

Factorization techniques involve breaking down numbers into a product of integers that can be multiplied together to obtain the original number. It simplifies complex problems and is extensively used in algebra.
Here is a step-by-step guide to factorization:

• Identify all factors: List out all factors of the number, which are integers that divide the number without leaving a remainder.
• Find perfect squares: Check which of the factors are perfect squares, as they simplify the expression.
• Pair and multiply: Form pairs with one number as a perfect square, such as 9 in the example of 45, enabling an answer like \(9 \cdot 5\).

Mastering these techniques makes it easier to tackle complex algebra problems efficiently.

Solving Algebra Problems with Factorization

Algebra problems often require the use of factorization to simplify equations or expressions. Recognizing perfect squares and forming integer pairs is an essential part of this process.
Let’s look at an example with the equation \(x^2 - 16 = 0\). By recognizing that 16 is \(4^2\), we can factorize this as \((x-4)(x+4) = 0\), solving for the roots \(x = 4\) and \(x = -4\).
Such techniques simplify the problem-solving process in algebra:

• Identify perfect squares within terms.
• Use factor pairs to break down and solve equations.
• Apply factorization to create simpler forms of equations, making them more manageable.

Understanding these concepts empowers students to approach various algebra problems with confidence.