Problem 69 - Algebra for College Students

Question

Solve each of the equations. $$a^{2}=5 a$$

Step-by-Step Solution

Answer

The solutions are \( a = 0 \) and \( a = 5 \).

Step 1: Set the Equation to Zero

Start by moving all terms to one side of the equation to set it to zero. This gives us: \[ a^2 - 5a = 0 \]

Step 2: Factor the Equation

Factor the equation by taking the common factor of \(a\) out of each term. The factored form of the equation is:\[ a(a - 5) = 0 \]

Step 3: Apply the Zero Product Property

Use the zero product property, which states that if a product of two terms equals zero, then at least one of the terms must be zero. So, set each factor to zero:1. \( a = 0 \)2. \( a - 5 = 0 \)

Step 4: Solve for 'a' from Each Factor

Solve each equation from the previous step:1. From \( a = 0 \), we have \( a = 0 \).2. From \( a - 5 = 0 \), solve for \( a \) by adding 5 to both sides, resulting in \( a = 5 \).

Key Concepts

Zero Product Property

When dealing with quadratic equations, the Zero Product Property is an essential concept. It states that if the product of two numbers is zero, then at least one of the two numbers must be zero. This property is useful in solving quadratic equations that have been factored into the form \[ (x - r)(x - s) = 0 \]Here, at least one of the factors \( (x - r) \) or \( (x - s) \) must equal zero to satisfy the equation:

• Set \( (x - r) = 0 \) to find one potential solution \( x = r \)
• Set \( (x - s) = 0 \) to find another potential solution \( x = s \)

This technique allows us to break down complex quadratic equations into simpler linear equations that are easy to solve. Each solution represents a value of the variable that makes the original quadratic equation true.

Factoring

Factoring is a method used to simplify quadratic equations, making them easier to solve by breaking them into their component terms. Consider the equation \( a^2 - 5a = 0 \). The goal is to express this equation as the product of its factors.To factor \( a^2 - 5a \), we look for common factors among the terms:

• Both terms, \( a^2 \) and \( 5a \), have \( a \) as a common factor.
• By factoring out \( a \), we rewrite the equation as \( a(a - 5) = 0 \).

This factored form reveals the solutions to the equation through the application of the Zero Product Property. Factoring transforms complex quadratic expressions into simple linear factors, paving the way for easy solution derivation.

Setting Equations to Zero

To effectively use the methods of factoring and the Zero Product Property, it's crucial to first set the equation to zero. This preparation step involves rearranging the equation to have zero on one side. For example, starting with \( a^2 = 5a \), we need to collect all terms on one side, transforming it into \( a^2 - 5a = 0 \).This step is essential because it

• Allows for factoring, which simplifies the equation into more manageable components.
• Sets the stage for using the Zero Product Property to find solutions.

By aligning all equation terms to one side, the equation becomes standardized, facilitating straightforward factorization. This standard form is a prerequisite for solving quadratic equations efficiently.