Problem 76 - Algebra for College Students

Question

Solve each of the equations. $$3 x=11 x^{2}$$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = \frac{3}{11} \).

Step 1: Rearrange Equation

Start by rearranging the equation to set it to zero: \[ 3x - 11x^2 = 0 \].This allows us to factor the equation, which is a common step when solving quadratic equations.

Step 2: Factor the Equation

Factor out the common term in the equation:\[ x(3 - 11x) = 0 \].This form will allow us to use the zero product property to find the solutions.

Step 3: Apply the Zero Product Property

Apply the zero product property, which states that if a product is zero, then at least one of the factors must be zero. So, set each factor equal to zero:1. \[ x = 0 \]2. \[ 3 - 11x = 0 \].

Step 4: Solve for x from the Second Factor

Solve the equation \[ 3 - 11x = 0 \] for \( x \):Add \( 11x \) to both sides:\[ 3 = 11x \]Divide both sides by 11:\[ x = \frac{3}{11} \].

Step 5: Present the Solutions

The solutions to the original equation \( 3x = 11x^2 \) are \[ x = 0 \] and \[ x = \frac{3}{11} \].

Key Concepts

Factoring Quadratic Equations

When you come across a quadratic equation, an essential skill is factoring it, especially when it is set to equal zero. Factoring is the process of breaking down an equation into simpler components, called "factors," which can be multiplied together to re-form the original equation. Consider the quadratic equation that has been rearranged: \( 3x - 11x^2 = 0 \). In this equation, we notice that both terms share a common factor: \( x \).
By factoring out \( x \), we write the equation as \( x(3 - 11x) = 0 \). This step makes it easier to solve by simplifying it into a product of terms. Factoring helps us because it gives us the opportunity to apply further techniques, like the Zero Product Property, to find the solutions of the equation.
To effectively factor equations, look for common variables or numbers and try to rewrite the equation in a simpler form.

Explaining the Zero Product Property

The Zero Product Property is a fundamental principle in algebra. It states that if the product of two factors is zero, then at least one of the factors must be zero. Let's see how it works in our context.
Returning to our factored equation: \( x(3 - 11x) = 0 \). According to the Zero Product Property:

• If \( x = 0 \), then \( x(3 - 11x) = 0 \).
• If \( 3 - 11x = 0 \), then \( x(3 - 11x) = 0 \).

Using this property, each factor can independently equal zero to satisfy the equation. This step is crucial as it transforms the problem into two simpler equations that can be tackled separately.
By setting each of these factors equal to zero, you are creating simpler equations that are much easier to solve, thus leading to the solutions of the original problem.

Solving Equations Using Factoring

Once we have applied the Zero Product Property to our factored equation, we are left with two straightforward equations to solve. Let's focus on these steps to solve the equation \( 3x = 11x^2 \):

• The first equation was \( x = 0 \), which is straightforward and one of the solutions.
• For the second equation, \( 3 - 11x = 0 \), solve for \( x \) by moving terms around and isolating \( x \). First, we rewrite it as \( 3 = 11x \) by adding \( 11x \) to both sides.
• Dividing both sides by 11 gives \( x = \frac{3}{11} \).

These solutions, \( x = 0 \) and \( x = \frac{3}{11} \), satisfy the original equation \( 3x = 11x^2 \). This sequence of steps demonstrates how to methodically approach solving a quadratic equation by using factoring and the Zero Product Property. Each step systematically reduces the complexity of the problem, making it easier to find the final solutions.