Problem 90 - Algebra A Combined Function

Question

Fill in the chart by finding two numbers that have the given product and sum. The first column is filled in for you. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline & & \text { 85. } & \text { 86. } & \text { 87. } & \text { 88. } & \text { 89. } & \text { 90. } & \text { 91. } & \text { 92. } \\ \hline \text { Two Numbers } & 4,7 & & & & & & & & \\ \hline \text { Their Product } & 28 & 12 & 20 & 8 & 16 & -10 & -9 & -24 & -36 \\\ \hline \text { Their Sum } & 11 & 8 & 9 & -9 & -10 & 3 & 0 & -5 & -5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

85) (2, 6); 86) (4, 5); 87) (-1, -8); 88) (-2, -8); 89) (5, -2); 90) (3, -3); 91) (3, -8); 92) (4, -9).

Step 1: Understand the Problem

We're given a table where we need to find two numbers for each row such that their product equals a given value and their sum equals another given value.

Step 2: Solve for Product 12 and Sum 8

We need two numbers, say \( x \) and \( y \), such that \( x \times y = 12 \) and \( x + y = 8 \). Start by listing factor pairs of 12: (1, 12), (2, 6), (3, 4). Check which pair sums to 8. The pair (2, 6) satisfies this condition.

Step 3: Solve for Product 20 and Sum 9

We need two numbers where \( x \times y = 20 \) and \( x + y = 9 \). Factor pairs of 20 are: (1, 20), (2, 10), (4, 5). The pair (4, 5) adds up to 9.

Step 4: Solve for Product 8 and Sum -9

We need numbers for \( x \times y = 8 \) and \( x + y = -9 \). Testing factor pairs of 8: (1, 8), (-1, -8), (2, 4), (-2, -4). The pair (-1, -8) sums to -9.

Step 5: Solve for Product 16 and Sum -10

Given \( x \times y = 16 \) and \( x + y = -10 \). Check factor pairs of 16: (1, 16), (-1, -16), (2, 8), (-2, -8). The pair (-2, -8) satisfies these conditions.

Step 6: Solve for Product -10 and Sum 3

We need \( x \times y = -10 \) and \( x + y = 3 \). Factor pairs of -10: (1, -10), (-1, 10), (2, -5), (-2, 5). The pair (5, -2) sums to 3.

Step 7: Solve for Product -9 and Sum 0

Required is \( x \times y = -9 \) and \( x + y = 0 \). Factor pairs of -9: (3, -3), (-3, 3). The pair (3, -3) adds to 0.

Step 8: Solve for Product -24 and Sum -5

For \( x \times y = -24 \) and \( x + y = -5 \), try factor pairs of -24: (1, -24), (-1, 24), (2, -12), (-2, 12), (3, -8), (-3, 8), (4, -6), (-4, 6). The pair (3, -8) adds to -5.

Step 9: Solve for Product -36 and Sum -5

We need \( x \times y = -36 \) and \( x + y = -5 \). Possible factor pairs: (1, -36), (-1, 36), (2, -18), (-2, 18), (3, -12), (-3, 12), (4, -9), (-4, 9), (6, -6), (-6, 6). The pair (4, -9) meets the requirement.

Key Concepts

Factor Pairs

When we talk about factor pairs in algebra, we are referring to two numbers that multiply together to give a specific product. Understanding factor pairs is crucial when solving problems involving the multiplication of two unknown numbers. For example, if we're looking for numbers with a product of 12, possible factor pairs are (1, 12), (2, 6), and (3, 4). Each of these pairs, when multiplied together, equals 12.
These factor pairs help us guess and check possibilities that may also satisfy other conditions, such as a specific sum. By systematically listing and testing these pairs, we can identify the one pair that meets all provided conditions in a problem.
It's a targeted approach that narrows down potential solutions, making it easier to find the correct answer efficiently.

Number Theory

Number theory is a branch of mathematics dealing with integers and their properties. It's the basis for understanding factorization, divisibility, and other relations between numbers. In solving algebraic problems involving sums and products, number theory provides essential insights.
By recognizing patterns and applying specific theorems, we can work efficiently with numbers to arrive at solutions. In our problem, for instance, number theory aids us in identifying factor pairs quickly and understanding their implications.
This area of mathematics empowers us with tools like the fundamental theorem of arithmetic, which states every integer greater than 1 can be represented uniquely as a product of prime numbers. Such principles make it easier to manage complex algebraic problems.

Sum and Product of Numbers

The sum and product of numbers are two fundamental operations in algebra. They are regularly used in problems where we need to find unknown numbers given these two operations' results. In our context, the sum is simply the result of adding two numbers, while the product results from multiplying them.
To solve these kinds of problems, we often look for two numbers such that both their sum and product meet specified conditions. This is why knowing the factor pairs is vital, as they give us a candidate pool for potential solutions.
By comparing the factor pairs with the required sum, we can pinpoint the exact numbers needed for the solution, making algebraic problems more approachable.