Problem 40 - Introductory Algebra for College Students

Question

Solve each equation by the method of your choice. Simplify irrational solutions, if possib $$x^{2}-6 x=0$$

Step-by-Step Solution

Answer

The solutions to the equation are \(x = 0\) and \(x = 6\).

Step 1: Factor out the variable x

The equation given is \(x^{2}-6 x=0\). By factoring out the \(x\) we have \(x(x - 6) = 0\)

Step 2: Solve for x using the Zero-Product Property

The Zero-Product Property states that if the product of two factors is zero, then at least one of the factors must be zero. Thus, we can set each factor of the equation equal to zero to solve for \(x\). So the solutions are \(x=0\) and \(x = 6\).

Key Concepts

Zero-Product Property

When solving quadratic equations, the Zero-Product Property is an essential tool. This property tells us that if the product of two numbers is zero, then at least one of those numbers must also be zero. It might seem straightforward, but it’s a powerful concept when it comes to solving equations.
In our context, once you have factored a quadratic equation, you break it down into simpler factors whose product equals zero. For instance, if you have an equation such as \( x(x - 6) = 0 \), according to the Zero-Product Property, you set each factor equal to zero separately. So, you would solve \( x = 0 \) and \( x - 6 = 0 \). This process allows you to find all possible solutions to the original equation, revealing the values of the variable where the product equals zero.

• This property applies strictly when the product equals zero—no other number!
• It is incredibly useful with factored forms of equations.
• It helps in breaking down complex equations into simpler ones to easily find solutions.

Factoring

Factoring is a key method used to simplify quadratic equations. It involves expressing the equation as the product of its simpler components or factors. In our example, we used factoring on the equation \( x^{2} - 6x = 0 \). Factoring helps in breaking down the equation into manageable parts, ultimately making it easier to solve.
In this step, we take the expression \( x^{2} - 6x \) and recognize that \( x \) is a common factor in both terms. So, we pull out the \( x \) from the equation, transforming it to \( x(x - 6) = 0 \). This formulation shows that the equation can now be tackled using simpler mathematical operations.
Effective factoring often requires:

• Identifying the greatest common factor of the terms.
• Recognizing patterns, such as the difference of squares or trinomial squares.
• Rewriting the equation in a factored form, making it suitable for solving.

Factoring is essential for utilizing the Zero-Product Property because it allows you to isolate the potential solutions in an equation.

Solving Equations

Solving equations, particularly quadratic ones, is a fundamental skill in mathematics. It involves finding the values of the variables that make the equation true. After factoring and applying the Zero-Product Property, you solve the individual simple equations to find these values.
Once the original equation \( x^{2} - 6x = 0 \) has been factored into \( x(x - 6) = 0 \), each part can be solved independently. For \( x = 0 \), it’s clear the solution is directly zero. For \( x - 6 = 0 \), you add 6 to both sides to find \( x = 6 \). This results in two solutions: \( x = 0 \) and \( x = 6 \).
Key points in solving equations are:

• Understanding each step in isolation.
• Being thorough with algebraic manipulations.
• Checking solutions by plugging them back into the original equation.

By following these approaches, you ensure that the solutions derived are correct and consistent with the equation given. While solving equations can seem daunting, breaking it down step by step simplifies the process greatly.