Problem 87 - Algebra for College Students

Question

Set up an equation and solve each of the following problems. The square of a number equals seven times the number. Find the number.

Step-by-Step Solution

Answer

The numbers are 0 and 7.

Step 1: Understanding the Problem

We are asked to find a number such that when the number is squared, it equals seven times the original number. This means if the number is represented as \( x \), the relationship can be expressed as \( x^2 = 7x \).

Step 2: Setting Up the Equation

Based on the problem description, we have the equation: \( x^2 = 7x \). Our goal is to find the values of \( x \) that satisfy this equation.

Step 3: Rewriting the Equation

To solve for \( x \), it's easier to set the equation to zero: \( x^2 - 7x = 0 \). This allows us to use factoring to find the solutions.

Step 4: Factoring the Equation

Factor out the common factor \( x \) from \( x^2 - 7x = 0 \). This gives: \( x(x - 7) = 0 \).

Step 5: Solving the Factored Equation

According to the zero-product property, for \( x(x - 7) = 0 \), \( x = 0 \) or \( x - 7 = 0 \). Solving \( x - 7 = 0 \) gives \( x = 7 \).

Step 6: Identifying the Solutions

We identified two potential solutions: \( x = 0 \) and \( x = 7 \). Both satisfy the original equation \( x^2 = 7x \).

Key Concepts

Solving Equations

When we talk about solving equations, it means finding the value of a variable that makes the equation true. In our exercise, the equation is given as \( x^2 = 7x \). This is a quadratic equation because it involves terms like \( x^2 \).
To solve such equations efficiently, we usually begin by rewriting the equation in standard form. This means getting all terms on one side of the equation equal to zero, so the equation becomes \( x^2 - 7x = 0 \).
Why do we do this? The standard form \( ax^2 + bx + c = 0 \) makes it easier to apply various solving techniques, such as factoring or using the quadratic formula. Remember:

• The goal is to find the values of \( x \) where the equation holds true.
• It often involves manipulating the equation and re-arranging terms.
• Other methods include graphing, completing the square, and more complex formulas.

Always check your solutions by substituting them back into the original equation.

Factoring

Factoring is a crucial method for solving quadratic equations like \( x^2 - 7x = 0 \). You might think of it like breaking down numbers into their simplest parts, like 6 into 2 and 3. Here, instead of numbers, we are dealing with expressions.
In our equation, observe the terms have a common factor, which is \( x \). By factoring, we rewrite the equation as \( x(x - 7) = 0 \).
Why is factoring useful? Because it transforms multiplication into a simpler form where we can easily identify solutions. Here's what you usually do during factoring:

• Look for common factors in terms.
• If possible, rewrite the equation as a product of simpler expressions.
• Once factored, employ the zero-product property to find solutions.

For beginners, factoring may seem daunting, but practice will make it second nature. It's a fast and efficient way to solve equations without resorting to more complex methods.

Zero-Product Property

Once you have factored your equation, the zero-product property becomes a powerful tool. This property states that if the product of two numbers equals zero, then at least one of the numbers must be zero.

In our exercise, after factoring the equation to \( x(x - 7) = 0 \), we apply this property:

• Either \( x = 0 \), which gives us one solution.
• Or \( x - 7 = 0 \), solve for \( x \) to find the second solution, \( x = 7 \).

The beauty of the zero-product property lies in its simplicity, allowing for straightforward computation of solutions after factoring. Once you master using this property, solving quadratic equations becomes much easier.

Always remember to check your solutions in the original equation to ensure they satisfy it. This confirmation step is key to validating your work.