Problem 91 - 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

The numbers are: 86: (2,6), 87: (4,5), 88: (-1,-8), 89: (-2,-8), 90: (-2,5), 91: (3,-3), and 92: (-8,3).

Step 1: Understanding the Problem

We are given a table with products and sums of two numbers. Our task is to identify the two numbers whose product and sum match the given values in the table for each exercise from 86 to 92. Column 85 is filled in as an example (4,7).

Step 2: Solving for Exercise 86

For exercise 86, we need two numbers that have a product of 12 and a sum of 8. The numbers 2 and 6 satisfy this condition since \(2 \times 6 = 12\) and \(2 + 6 = 8\).

Step 3: Solving for Exercise 87

For exercise 87, we need two numbers that have a product of 20 and a sum of 9. The numbers 4 and 5 satisfy this condition since \(4 \times 5 = 20\) and \(4 + 5 = 9\).

Step 4: Solving for Exercise 88

For exercise 88, we need two numbers that have a product of 8 and a sum of -9. The numbers -1 and -8 satisfy this condition since \(-1 \times (-8) = 8\) and \(-1 + (-8) = -9\).

Step 5: Solving for Exercise 89

For exercise 89, we need two numbers that have a product of 16 and a sum of -10. The numbers -2 and -8 satisfy this condition since \(-2 \times (-8) = 16\) and \(-2 + (-8) = -10\).

Step 6: Solving for Exercise 90

For exercise 90, we need two numbers that have a product of -10 and a sum of 3. The numbers -2 and 5 satisfy this condition since \(-2 \times 5 = -10\) and \(-2 + 5 = 3\).

Step 7: Solving for Exercise 91

For exercise 91, we need two numbers that have a product of -9 and a sum of 0. The numbers 3 and -3 satisfy this condition since \(3 \times (-3) = -9\) and \(3 + (-3) = 0\).

Step 8: Solving for Exercise 92

For exercise 92, we need two numbers that have a product of -24 and a sum of -5. The numbers -8 and 3 satisfy this condition since \(-8 \times 3 = -24\) and \(-8 + 3 = -5\).

Key Concepts

Finding Two Numbers

Finding two numbers that meet specific conditions is a common type of word problem in algebra. The main goal is to determine two numbers that, when interacted through multiplication and addition, result in the given product and sum, respectively.

• Consider the problem requirements carefully. You need one pair of numbers for each given condition.


• Identify any pairs of factors that multiply together to give the product. Not every combination will work because we also need to check the sum condition.


• Test each pair against the sum condition. Only the pair that meets both the product and sum conditions is the correct answer.

Understanding this dual requirement is key in approaching and solving such problems efficiently. The conditions provided in exercises often help guide your path to the solution.

Product and Sum

The product and sum of two numbers provide a unique combination of conditions that help in determining the numbers themselves. Here's how you can interpret these conditions:

• **Product**: This is the result when two numbers are multiplied together. In algebra, exploring factor combinations gives multiple possibilities.


• **Sum**: This is the result when the same two numbers are added. The sum condition narrows down which factor pairs are valid, as not all pairs that multiply to the product will sum correctly.

One effective strategy is to list all possible pairs of factors of the product. Then, check which of these pairs sum to give the necessary total. This second condition helps eliminate incorrect pairs and leads directly to the solution.

Step by Step Solution

Approaching this type of problem with a step-by-step methodical solution is helpful for understanding and accuracy. Here’s a breakdown of how such solutions typically unfold:

• Read the problem carefully and note down the given conditions: the product and the sum.


• List factor pairs of the product. For each exercise, the multiplication must give the product specified.


• Calculate the sum of each factor pair. Check to see if it matches the sum given in the problem.


• Verify each possibility before confirming the solution. Double-check to ensure that both conditions are met.

This procedure resembles a logical flow, aiding in systematically isolating the correct numeric pair from a potential list of candidates.

Integer Solutions

In algebra word problems involving products and sums, solutions must often be integers. The restrictions of integer solutions simplify some aspects but can also complicate others.

• An **integer** is a whole number, positive, negative, or zero. No fractions or decimals are allowed.


• This restriction typically limits the possibilities compared to if any real numbers were allowed.


• Being integers usually simplifies calculations, since integers have clear multiplication and addition results.

When integrating these constraints in word problems, it is important to ensure number choices are whole numbers, and they satisfy all conditions set out initially. This potential simplification helps streamline the solving process, focusing on only viable whole number options.