Problem 53 - Algebra A Combined Function

Question

Fill in each blank. $$ 36 x^{2}=4 x $$ ________

Step-by-Step Solution

Answer

\(x = 0\) and \(x = \frac{1}{9}\).

Step 1: Bring All Terms to One Side of the Equation

Start by bringing all the terms to one side of the equation to set it to zero:\[36x^2 - 4x = 0\]This will allow us to use factoring to solve for \(x\).

Step 2: Factor Out the Greatest Common Factor

Identify and factor out the greatest common factor (GCF) from the equation. In this case, the GCF is \(4x\):\[4x(9x - 1) = 0\]This factors the equation into two terms that multiply to zero.

Step 3: Apply the Zero Product Property

The Zero Product Property states that if a product of factors is zero, at least one of the factors must be zero. We can apply this property to set each factor equal to zero:\[4x = 0 \quad \text{or} \quad 9x - 1 = 0\]

Step 4: Solve Each Equation Separately

Solve each equation from the previous step:1. Solve \(4x = 0\): \[x = 0\]2. Solve \(9x - 1 = 0\): \[9x = 1 \x = \frac{1}{9}\]

Step 5: Conclusion of Solutions

The solutions to the equation are \(x = 0\) and \(x = \frac{1}{9}\). Therefore, the blanks can be filled with these values.

Key Concepts

Zero Product Property

The Zero Product Property is a fundamental principle in algebra that helps us solve equations that have been factored into a product of terms. This property tells us that if the product of two or more factors equals zero, then at least one of the factors must be zero.
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This makes sense because multiplying any number by zero results in zero. Let's consider the equation from the exercise:

• Factored form: \(4x(9x - 1) = 0\)

According to the Zero Product Property, either one of these must be zero:

• \(4x = 0\)
• \(9x - 1 = 0\)

Therefore, solving each factor independently allows us to find the possible solutions for \(x\). This method is especially useful in solving quadratic equations because it simplifies the problem into smaller, easier segments.

Greatest Common Factor

Finding the Greatest Common Factor (GCF) is an essential step when you're preparing to solve equations through factoring. The GCF is the largest factor that divides two or more numbers. In the problem, factoring out the GCF helps simplify the equation and makes it easier to apply subsequent solution steps.
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Let's see this in action with our problem:

• Original equation: \(36x^2 - 4x = 0\)
• Identify the GCF: Notice that both terms \(36x^2\) and \(4x\) contain \(4x\)
• Factor the GCF out of the equation: \(4x(9x - 1) = 0\)

By factoring out \(4x\), the equation is simplified, leaving us with a straightforward expression that can be easily solved using the Zero Product Property. This not only streamlines the solving process but also reduces the risk of errors.

Solving Quadratic Equations

Solving quadratic equations is a critical skill in algebra, allowing us to find the values of \(x\) that make the equation true. Quadratic equations are generally in the form \(ax^2 + bx + c = 0\), and their solutions can often be identified using a variety of methods such as factoring, the quadratic formula, or completing the square.
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For the equation \(36x^2 - 4x = 0\), the factorization method is used:

• Rewritten form: \(4x(9x - 1) = 0\)
• Use the Zero Product Property, suggesting two separate equations to solve: \(4x = 0\) and \(9x - 1 = 0\)
• Solve each equation for \(x\):
  • \(4x = 0\) gives \(x = 0\)
  • \(9x - 1 = 0\) solves to \(x = \frac{1}{9}\)

Both solutions, \(x = 0\) and \(x = \frac{1}{9}\), satisfy the original equation. This method demonstrates the power of factoring and how breaking down complex problems into simpler parts can yield an effective solution.