Problem 46 - Elementary and Intermediate Algebra

Question

Solve each equation. $$ p^{2}=20 p $$

Step-by-Step Solution

Answer

The solutions are \( p = 0 \) and \( p = 20 \).

Step 1: Rearrange the Equation

Given the equation \( p^2 = 20p \), the first step is to rearrange it into a standard quadratic form. Subtract \( 20p \) from both sides to get: \( p^2 - 20p = 0 \).

Step 2: Factor the Equation

In the equation \( p^2 - 20p = 0 \), notice that you can factor out a common factor \( p \) from each term. This gives us \( p(p - 20) = 0 \).

Step 3: Set Each Factor to Zero

For the equation \( p(p - 20) = 0 \) to be true, either \( p = 0 \) or \( p - 20 = 0 \). These are two separate equations to solve.

Step 4: Solve the Simple Equations

Solve each factor separately: - For \( p = 0 \), the solution is \( p = 0 \). - For \( p - 20 = 0 \), add 20 to both sides to solve for \( p \), giving \( p = 20 \).

Key Concepts

Factoring

Factoring is a crucial step in solving many types of equations, especially quadratic equations. It involves expressing an equation as a product of its component factors. Consider the equation you receive after rearranging:

• From: \( p^2 - 20p = 0 \)
• To: \( p(p - 20) = 0 \)

In this expression, the common factor is \( p \). By factoring it out, you break the equation into simpler parts: \( p \) and \( p - 20 \). These factors allow you to work with more straightforward equations. Factoring helps reduce complexity by transforming a single complex equation into manageable bite-sized pieces.
Recognizing common factors is key. Practice regularly with different equations to hone your factoring skills and make this process second nature.

Zero Product Property

The Zero Product Property is a powerful principle in algebra. It states that if a product of two factors equals zero, at least one of the factors must be zero.

• For an equation like \( p(p - 20) = 0 \), either \( p = 0 \) or \( p - 20 = 0 \).

This property is your shortcut to finding solutions to factored equations. It allows you to set each factor in the product equal to zero, creating individual, simpler equations to solve. This principle is highly effective because solving these smaller problems is generally much easier than tackling the original equation directly.
Understanding and applying the Zero Product Property will streamline your approach to solving quadratics and save valuable time. Master it by applying this rule repeatedly in diverse quadratic exercises.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations to make them easier to solve. In the step-by-step solution provided:

• Starting with: \( p^2 = 20p \)
• Rearranged to: \( p^2 - 20p = 0 \)

subtracting \( 20p \) from both sides places the equation into a standard quadratic form. This step is essential because working in standard form allows for consistent solving strategies, such as factoring.
By practicing algebraic manipulations, you'll gain fluency in recognizing which operations will simplify an equation. This skill underpins your ability to transform equations, isolate variables, and ultimately reach solutions efficiently. Practice making these transformations confidently to refine how you approach problems in algebra.