Problem 68 - Algebra for College Students

Question

Solve each of the equations. $$x^{2}-14 x=0$$

Step-by-Step Solution

Answer

The solutions are \( x = 0 \) and \( x = 14 \).

Step 1: Factor the Equation

The given equation is a quadratic equation: \[ x^2 - 14x = 0 \] The first step is to factor the equation. Notice that you can factor an \(x\) from each term: \[ x(x - 14) = 0 \]

Step 2: Apply the Zero-Product Property

The Zero-Product Property states that if a product of two factors is zero, then at least one of the factors must be zero. We apply this property to our factored equation:- Set each factor equal to zero: \[ x = 0 \] \[ x - 14 = 0 \]

Step 3: Solve Each Equation

Solve each of the simple equations obtained from the factored form:- The first equation \( x = 0 \) is already solved.- Solve the second equation for \( x \): \[ x - 14 = 0 \] Add 14 to both sides: \[ x = 14 \]

Step 4: State the Solutions

The solutions to the original equation \( x^2 - 14x = 0 \) are the values of \( x \) that satisfy both simple equations. Therefore, the solutions are:- \( x = 0 \)- \( x = 14 \)

Key Concepts

Factoring Method

The factoring method is a fundamental technique used to solve quadratic equations, typically expressed in the form \( ax^2 + bx + c = 0 \). This method involves rewriting the quadratic equation as a product of its factors. In other words, you transform a polynomial into a product of simpler expressions. This transformation is crucial because it lays the groundwork for applying other solving methods, such as the zero-product property.

Here's how it typically works:

• Identify common factors in each term of the quadratic equation. In equations like \( x^2 - 14x = 0 \), you can see that an \( x \) is common in both terms.
• Factor out the common term and rewrite the expression as a product. For our equation, factoring yields \( x(x - 14) = 0 \).

This step simplifies the original equation and sets the stage for finding solutions by making it easier to apply the zero-product property.

Zero-Product Property

The zero-product property is a simple but powerful concept that states: if the product of two numbers is zero, then at least one of those numbers must be zero. In mathematical terms, if \( a \times b = 0 \), then either \( a = 0 \) or \( b = 0 \). This property allows us to break down complex equations into simpler pieces that are easier to solve.

When we apply this property to a factored quadratic equation such as \( x(x - 14) = 0 \), we obtain two straightforward equations to solve:

• \( x = 0 \)
• \( x - 14 = 0 \)

By solving each equation individually, you ensure that you capture all possible solutions to the original quadratic equation.

Solving Quadratic Equations

Solving quadratic equations is an essential skill in algebra, involving finding the values of the variable that satisfy the equation. Once your quadratic equation is in a factored form, as seen in equations like \( x(x - 14) = 0 \), you can easily apply the zero-product property to find the solutions.

Let's go through the example:

• Start by solving each equation from the factored form separately. The equation \( x = 0 \) is already solved.
• For the equation \( x - 14 = 0 \), add 14 to both sides: \( x = 14 \).

Thus, the solutions to the original quadratic equation \( x^2 - 14x = 0 \) are \( x = 0 \) and \( x = 14 \). These solutions represent the points where the graph of the equation would intersect the x-axis, known as the roots or zeros of the equation.