Problem 78 - Algebra for College Students

Question

Solve each of the equations. $$x-6 x^{2}=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = \frac{1}{6} \).

Step 1: Identify Common Factor

Look at the equation \( x - 6x^2 = 0 \). Identify the common factor in both terms. Both terms have a common factor of \( x \).

Step 2: Factor Out the Common Factor

Factor \( x \) out of the expression, so the equation becomes: \[ x(1 - 6x) = 0 \]

Step 3: Apply the Zero Product Property

Use the zero product property, which says if a product of two numbers is zero, then at least one of the numbers must be zero. So, set each factor equal to zero: 1. \( x = 0 \) 2. \( 1 - 6x = 0 \)

Step 4: Solve Each Equation

Solve each of the equations from Step 3: 1. For \( x = 0 \), the solution is simply \( x = 0 \). 2. Solve \( 1 - 6x = 0 \) by adding \( 6x \) to both sides to get \( 1 = 6x \), then divide by 6: \[ x = \frac{1}{6} \]

Step 5: Verify Solutions

Check both solutions in the original equation: 1. For \( x = 0 \): \( 0 - 6(0)^2 = 0 \). This is true. 2. For \( x = \frac{1}{6} \): \( \frac{1}{6} - 6\left(\frac{1}{6}\right)^2 = 0 \). Simplifying gives \( \frac{1}{6} - \frac{1}{6} = 0 \), which is true.

Key Concepts

Factorization

Factorization is a method used in algebra to simplify equations, making them easier to solve. When faced with a quadratic equation like \( x - 6x^2 = 0 \), the first step is to identify a common factor in all the terms. In this case, both terms share a common factor of \( x \).

Once a common factor is identified, you can "factor it out". This means rewriting the equation by taking the common factor outside of a parenthesis. For our equation, it becomes \( x(1 - 6x) = 0 \).

By doing this, we transform the equation into a product of two factors, which will be crucial in the next step. This process simplifies the equation structure and sets up further steps to find the solutions easily.

Zero Product Property

The zero product property is a powerful tool in solving algebraic equations that involve multiplication. It states that if the product of two numbers is zero, then at least one of the numbers must itself be zero. This principle is what we use after factorizing an equation into a simple product form.

For the equation \( x(1 - 6x) = 0 \), applying the zero product property allows us to set each factor independently equal to zero:

• \( x = 0 \)
• \( 1 - 6x = 0 \)

This step is crucial because it separates the problem into smaller, often simpler parts that can be solved individually to find all possible solutions. Ensuring each component of the product is examined ensures no solutions are overlooked.

Checking Solutions

Checking solutions is an important final step in solving equations to ensure that the found values actually satisfy the original equation. This step prevents errors and confirms the correctness of your solutions.

In the example equation \( x - 6x^2 = 0 \), suppose we found solutions \( x = 0 \) and \( x = \frac{1}{6} \). We substitute these back into the original equation to verify:

• For \( x = 0 \): Substitute and get \( 0 - 6(0)^2 = 0 \), which holds true.
• For \( x = \frac{1}{6} \): Substitute and simplify to \( \frac{1}{6} - \frac{1}{6} = 0 \), confirming that it also holds true.

This step not only confirms the logic applied in earlier steps but also boosts confidence in the calculated results.