Problem 24 - Algebra A Combined Function

Question

Solve. $$ x^{2}-3 x=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 0\) and \(x = 3\).

Step 1: Identify the Given Equation

The given equation is a quadratic equation of the form \(x^2 - 3x = 0\). This equation needs to be solved for \(x\).

Step 2: Factor the Equation

The equation \(x^2 - 3x = 0\) can be factored by taking \(x\) common from both terms, resulting in \(x(x - 3) = 0\).

Step 3: Apply the Zero Product Property

According to the zero product property, if \(ab = 0\), then either \(a = 0\) or \(b = 0\), or both. Therefore, set each factor of the equation \(x(x - 3) = 0\) to zero: \(x = 0\) or \(x - 3 = 0\).

Step 4: Solve Each Equation

Solve the two separate equations obtained from the zero product property:1. \(x = 0\) 2. \(x - 3 = 0\) which simplifies to \(x = 3\).

Step 5: Present the Solution

The solutions to the quadratic equation \(x^2 - 3x = 0\) are \(x = 0\) and \(x = 3\).

Key Concepts

Factoring Method

The factoring method is a powerful technique to solve quadratic equations. Quadratic equations generally take the form \(ax^2 + bx + c = 0\). When solving such equations via factoring, the goal is to express the equation as a product of two simpler expressions that, when multiplied, equate to zero.

In the given equation \(x^2 - 3x = 0\), observe how you can take \(x\) common from both terms, resulting in the factored form \(x(x - 3) = 0\). This is achieved by focusing on common factors in each term.

• The first term \(x^2\) breaks down to \(x \times x\).
• The second term \(-3x\) identifies \(x\) as a common element.

This process ensures that the quadratic expression is rewritten in a way that sets the stage for the zero product property to come into play.

Zero Product Property

The zero product property is a fundamental concept in algebra. It states that if a product of two terms equals zero, then at least one of the terms must be zero. This allows us to break down a factored quadratic equation into simpler linear equations.

In our example, after factoring, we have \(x(x - 3) = 0\). Applying the zero product property here means:

• Either \(x = 0\), or
• \(x - 3 = 0\)

By setting each factor equal to zero, we can easily find the possible values of \(x\). This property is particularly handy because it converts the problem into a series of straightforward linear equations, making it easier to identify all solutions.

Solving Quadratic Equations

Solving quadratic equations typically involves finding the values of a variable that satisfy the equation. Our example uses the factoring method combined with the zero product property. These steps together unravel the values of \(x\) for the given quadratic equation.

Once the equation \(x^2 - 3x = 0\) is factored and expressed as \(x(x - 3) = 0\), we apply the zero product property. Solving \(x = 0\) directly gives us a solution. Similarly, solving \(x - 3 = 0\) yields \(x = 3\).

Hence, the solutions to the quadratic equation are:

• \(x = 0\)
• \(x = 3\)

By following the method of factoring and utilizing the zero product property, we arrive at the final solutions efficiently. Remember, solving quadratic equations typically follows three main paths: factoring, using the quadratic formula, or completing the square, with factoring often being the simplest and most direct approach when it applies.