Problem 38 - Beginning and Intermediate Algebra

Question

Solve each equation. $$d^{2}=d$$

Step-by-Step Solution

Answer

The given equation is \(d^2 = d\). By subtracting d from both sides and factoring, we get \(d(d - 1) = 0\). Thus, the solutions for \(d\) are \(0\) and \(1\).

Step 1: Rewrite the equation

Rewrite the given equation: \(d^2 = d\).

Step 2: Move all terms to one side of the equation

Subtract \(d\) from both sides of the equation to move all terms to one side: \(d^2 - d = 0\).

Step 3: Factor the equation

Factor the equation \(d^2-d=0\) by taking out a common factor of \(d\): \[d(d - 1) = 0\]

Step 4: Solve for d

Since a product is equal to zero if and only if one of the factors is equal to zero, we can solve for \(d\) by setting each factor to zero: Case 1: \(d = 0\) Case 2: \(d - 1 = 0\), which can be solved by adding 1 to both sides to get: \(d = 1\). So, there are two possible solutions for \(d\) in the given equation: \(d = 0\) and \(d = 1\).

Key Concepts

Factoring

Factoring is a crucial tool when dealing with quadratic equations. In our exercise, we began by looking at the equation \(d^2 = d\). To solve the equation, the first step is to ensure all terms are on one side, resulting in \(d^2 - d = 0\).
This equation can be factored by identifying a common factor in each term, which in this case is \(d\).
This is done as follows:

• Identify the common factor in each term of the equation, which is the greatest common divisor - here, it's the variable \(d\).
• Extract the common factor and write the equation as a product: \(d(d - 1) = 0\).

The goal of factoring is to rewrite the equation as a product of simpler expressions that can be easily solved. Factoring transforms a complex-looking quadratic equation into an understandable form of the product.

Zero Product Property

The Zero Product Property is an essential concept that helps solve quadratic equations effectively.
This property states that if the product of two quantities equals zero, at least one of the quantities must be zero.
In our example, after factoring the equation into \(d(d - 1) = 0\), we apply this rule. This gives us two simple scenarios:

• Scenario 1: \(d = 0\)
• Scenario 2: \(d - 1 = 0\)

By setting each factor equal to zero separately, we gain insight into the possible solutions of the equation.
The Zero Product Property makes solving equations straightforward once they are factored, allowing us to break down potentially complex problems into more manageable parts.

Solving Equations

Solving equations follows naturally after factoring and using the Zero Product Property.
Once the equation \(d(d - 1) = 0\) is established, we proceed by tackling each part concurrently:

• For the factor \(d = 0\), we immediately recognize one solution, which is straightforward and requires no further steps.
• For the factor \(d - 1 = 0\), solving involves adding 1 to both sides, yielding \(d = 1\) as the second solution.

The resolute steps lead to two possible values of \(d\), confirming the solutions as \(d = 0\) and \(d = 1\).
Solving equations can sometimes involve multiple steps, such as isolating variables, using operations to simplify, and ensuring all solutions meet the original equation’s requirements.
These steps promote a clear and comprehensive approach toward finding every possible solution.