Problem 30 - Algebra for College Students

Question

Solve each equation for \(x\) by factoring and applying the property, \(a b=0\) if and only if \(a=0\) or \(b=0\). $$x^{2}=25 k^{2} x$$

Step-by-Step Solution

Answer

Solutions are \(x = 0\) and \(x = 25k^{2}\).

Step 1: Re-arrange Equation

First, we re-arrange the given equation from\[x^{2} = 25k^{2}x\]to standard form, which is\[x^{2} - 25k^{2}x = 0.\]This allows us to set the equation to zero, making it ready for factoring.

Step 2: Factor the Equation

Now, we factor out the common factor from\[x^{2} - 25k^{2}x = 0.\]The common factor here is\(x\), so we factor it out:\[x(x - 25k^{2}) = 0.\]This expresses the equation as a product of two terms.

Step 3: Apply the Zero Product Property

Apply the Zero Product Property which states that if \(a\cdot b = 0\), then \(a = 0\) or \(b = 0\). From\[x(x - 25k^{2}) = 0,\]we get two separate equations to solve: \(x = 0\) and \(x - 25k^{2} = 0.\)

Step 4: Solve the Equations

Solve each equation separately. The first one is simple: already \(x = 0\). The second equation \(x - 25k^{2} = 0\), solve it by adding \(25k^{2}\) to both sides:\[x = 25k^{2}.\]So, the solutions to the equation are \(x = 0\) and \(x = 25k^{2}.\)

Key Concepts

Zero Product Property

The zero product property is a foundational concept in algebra that plays a key role in solving equations. It states that if a product of two numbers (or variables) equals zero, then at least one of the multiplicands must be zero. This can be written as:

• If \(a \cdot b = 0\), then \(a = 0\) or \(b = 0\).

This property is powerful because it simplifies solving equations that can be factored into products. When you encounter a quadratic equation in factored form, you can immediately set each factor equal to zero and solve for the variable. This provides the solution to the equation quickly and efficiently. In our exercise, after factoring the equation into \(x(x - 25k^{2}) = 0\), we applied this property. Setting each factor to zero gives us the equations \(x = 0\) and \(x - 25k^{2} = 0\), which simplifies the process of finding solutions.

Solving Quadratic Equations

Solving quadratic equations is a central task in algebra, and there are several methods to achieve this, such as factoring, completing the square, and using the quadratic formula. The solution involves finding the values of \(x\) that make the equation true. Quadratic equations are often written in the standard form \(ax^2 + bx + c = 0\).

• In the example \(x^{2} - 25k^{2}x = 0\), the equation is already set to zero, which is an essential step for solving any quadratic equation.
• Next, recognizing that a common factor \(x\) can be factored out simplifies the equation into a product of two terms.
• Once factored, applying the zero product property allows solving the resulting linear equations for \(x\).

By converting the equation into its factored form, the quadratic equation solutions, \(x = 0\) and \(x = 25k^{2}\), emerge smoothly.

Factoring Techniques

Factoring is a key technique to simplify quadratic equations and make them solvable through the zero product property. It involves expressing an equation as a product of simpler terms. Several techniques exist:

• Common Factor Factoring: Identify and extract the greatest common factor (GCF) from all terms in the equation. In \(x^{2} - 25k^{2}x\), the GCF is \(x\).
• Grouping: Split the middle term to group terms into pairs and factor each group, though this is not applied in the given problem.
• Difference of Squares: Useful for equations like \(a^2 - b^2\), however, not used in the given exercise as it's linear after common factoring.

In our example, factoring the equation rendered it into two components: \(x\) and \(x - 25k^{2}\). With these techniques, any polynomial expression can become simpler to solve.