Problem 86 - 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

86: 2,6; 87: 4,5; 88: -1,-8; 89: -2,-8; 90: 5,-2; 91: 3,-3; 92: -8,3.

Step 1: Understanding the Problem

We have a table where each row lists two numbers, their product, and their sum. Our task is to identify the two numbers given their product and sum for questions 86 to 92.

Step 2: Solving for Question 86

For question 86, we need two numbers whose product is 12 and sum is 8. Let the numbers be \(x\) and \(y\). Thus, we have the equations: \(x \cdot y = 12\) and \(x + y = 8\). Solving these, the numbers are 2 and 6.

Step 3: Solving for Question 87

For question 87, we need two numbers that multiply to 20 and add up to 9. Solving the equations \(x \cdot y = 20\) and \(x + y = 9\), the numbers are 4 and 5.

Step 4: Solving for Question 88

Here, the problem asks for two numbers that multiply to 8 and sum to -9. Solving \(x \cdot y = 8\) and \(x + y = -9\), the numbers are -1 and -8.

Step 5: Solving for Question 89

In question 89, we need numbers with product 16 and sum -10. Using \(x \cdot y = 16\) and \(x + y = -10\) yields the numbers -2 and -8.

Step 6: Solving for Question 90

For question 90, we have a product of -10 and a sum of 3. Solving \(x \cdot y = -10\) and \(x + y = 3\), we find the numbers 5 and -2.

Step 7: Solving for Question 91

In this case, the product is -9 and the sum is 0. Using \(x \cdot y = -9\) and \(x + y = 0\), the numbers are 3 and -3.

Step 8: Solving for Question 92

For this question, the numbers should have a product of -24 and a sum of -5. Solving \(x \cdot y = -24\) and \(x + y = -5\), we get the numbers -8 and 3.

Key Concepts

Product and Sum Problems

Product and sum problems involve finding two numbers that meet specific criteria related to both their multiplication (product) and addition (sum). This task tests our understanding of how numbers relate to each other through these operations. For example, in problem-solving exercises, we may be given a product like 12 and a sum like 8, and we need to determine numbers that fit these conditions.

To tackle these problems, it's useful to set up a system of equations. Let's say we denote our numbers as \( x \) and \( y \). We'll generally have two equations:

• The product equation: \( x \cdot y = \text{Product} \)
• The sum equation: \( x + y = \text{Sum} \)

We solve this system to find possible pairs of numbers. These problems often appear in algebra courses and help students practice basic equation-solving skills, with an added layer of finding pairs that satisfy multiple conditions.

Solving Equations

Solving equations is a fundamental part of algebra. In the context of product and sum problems, solving a pair of simultaneous equations is key. When faced with a pair of equations, such as \( x \cdot y = 12 \) and \( x + y = 8 \), our goal is to find the values of \( x \) and \( y \).

Here are steps to solve such pairs of equations:

• Start by expressing one variable in terms of the other from the sum equation. For example, \( y = 8 - x \).
• Substitute this expression into the product equation, replacing \( y \), e.g., \( x \cdot (8 - x) = 12 \).
• This often results in a quadratic equation like \( x^2 - 8x + 12 = 0 \), which can be solved using factoring, the quadratic formula, or completing the square.
  
• Factor the quadratic expression, if possible, to find the roots. These roots represent the values of \( x \).
• Calculate \( y \) using one or both roots by substituting back into \( y = 8 - x \).

Through these steps, we determine the pair(s) of numbers that satisfy both the product and sum conditions.

Number Relationships

Understanding number relationships is crucial when solving product and sum problems. Relationships help us guess or predict possible solutions and verify them as well.

The product and sum tell us about how two numbers relates:

• The product gives insight into the multiplicative relationship, indicating whether the numbers are small/large, positive/negative, or identical.
• The sum, on the other hand, provides information about their additive relationship, showing a possible range or difference between the numbers.

For instance, if the product of two numbers is negative, we instantly know that one of the numbers must be positive and the other negative. Similarly, if their sum is zero, the numbers must be equal in magnitude but opposite in sign. Recognizing these patterns can make finding solutions faster and more intuitive, as they guide us towards setting up our equations effectively and predicting logical pairs of values. Number relationships are a valuable tool in breaking down complex algebraic problems into simpler tasks.