Solving Quadratic Equations: A Guide to Finding Numbers Given Their Product and Difference

Understanding how to solve quadratic equations and apply them to word problems is a fundamental skill in algebra. This guide will walk you through solving a specific problem: if one number is 5 greater than another and their product is 36, what are the numbers? We will explore the problem step by step, using clear mathematical reasoning and a systematic approach.

Problem Statement

The problem statement provides us with two key pieces of information:

One number is 5 greater than another. Their product is 36.

Let's denote the smaller number as x. Based on the first piece of information, the larger number can be expressed as x 5.

Formulating the Equation

We know that the product of these two numbers is 36. Hence, we can set up the following equation:

x(x 5) 36

Expanding this equation:

x^2 5x - 36 0

Solving the Quadratic Equation

Now that we have a quadratic equation, we can solve it using the quadratic formula:

x u00bd(-b u00B1 u221A(b^2 - 4ac))

Here, a 1, b 5, and c -36. Plugging these values into the formula:

x u00bd(-5 u00B1 u221A(5^2 - 4 cdot 1 cdot -36))

x u00bd(-5 u00B1 u221A(25 144))

x u00bd(-5 u00B1 u221A(169))

x u00bd(-5 u00B1 13)

This gives us two potential solutions:

x u00bd(8) 4

x u00bd(-18) -9

Identifying the Correct Solutions

Now that we have two potential solutions for x, we can find the corresponding larger number for each case:

If x 4, then the larger number is 4 5 9. If x -9, then the larger number is -9 5 -4.

There are two sets of numbers that satisfy the given conditions: (4, 9) and (-9, -4). However, the most common and logical interpretation would be the positive numbers:

The numbers are 4 and 9.

Additional Problem Solved for Reference

For further reference, let's solve another problem of the same nature, where the product is 1936 and one number is 4 times the other. Let x and y be the two numbers, where y 4x. Then, according to the conditions:

xy 1936 y 4x

Substituting y 4x into the first equation:

x(4x) 1936

This simplifies to:

4x^2 1936

Dividing both sides by 4:

x^2 484

Taking the square root of both sides:

x u221A484 22

Hence, the larger number is:

4x 4 u00D7 22 88

So the numbers are 22 and 88.