Problem 27 - Elementary Algebra

Question

Solve. $$ c(4-c)=0 $$

Step-by-Step Solution

Answer

The solutions are \( c = 0 \) and \( c = 4 \).

Step 1: Identify the Equation

The given equation is a quadratic equation in the form of a product of two factors: \( c(4 - c) = 0 \).

Step 2: Understand the Zero Product Property

According to the zero product property, if the product of two factors is zero, at least one of the factors must be zero. So, solve the equation by setting each factor equal to zero separately.

Step 3: Solve the First Factor

Set the first factor equal to zero: \( c = 0 \).

Step 4: Solve the Second Factor

Set the second factor equal to zero: \( 4 - c = 0 \). Rearrange to solve for \( c \): \( c = 4 \).

Step 5: List All Solutions

Combine the solutions from both factors: \( c = 0 \) and \( c = 4 \).

Key Concepts

Zero Product Property

Understanding the zero product property is key to solving many quadratic equations. This property states that when the product of two factors equals zero, at least one of the factors must be zero.
In simple terms, if you have an equation that looks like \(A \cdot B = 0\), then either \(A = 0\) or \(B = 0\), or both. This is a very useful concept when dealing with quadratics, as it helps break down complex equations into simpler parts.

Factoring Quadratics

Factoring quadratics involves rewriting the quadratic equation in a product form. For instance, the equation \( c(4-c)=0 \) is already factored. In this form, each part can be considered separately.
Here are key steps to factor more complex quadratics:

• Look for common factors
• Apply techniques like splitting the middle term
• Check if the quadratic is a special form like a perfect square

A factored quadratic will generally be easier to solve using the zero product property.

Solutions of Quadratic Equations

Solving quadratic equations involves finding the values of the variable that make the equation true. Using the zero product property, we can find these values from the equation.
Let's revisit our example: \( c(4-c)=0 \). By applying the zero product property, we set each factor equal to zero: \( c = 0 \) and \( 4 - c = 0 \).
Then solve for \( c \) in each case:

• \( c = 0 \)
• \( 4 - c = 0 \). Rearrange to get \( c = 4 \)

Therefore, the solutions to the quadratic equation are \( c = 0 \) and \( c = 4 \).