Problem 100 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. (Lesson 10.4) $$ 6(x-14)^{2}=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(6(x-14)^{2}=0\) is \(x = 14\).

Step 1: Apply Zero Product Property

Start by applying the zero-product property, which states that if a product of two factors is zero then at least one of the factors must be zero. We have two factors here, 6 and \((x-14)^{2}\). So we can set each one equal to zero.

Step 2: Solve the equation for \(x\)

We know 6 is not equal to zero so we can safely ignore it. For the second factor, we set \((x-14)^{2}=0\). If we take the square root of both sides, which is the inverse operation of squaring, we get \(x-14 = 0\). Solving for 'x' gives us \(x = 14\) as the solution.

Key Concepts

Quadratic Equations

Quadratic equations are polynomial equations of degree 2, and they typically take the form \(ax^2 + bx + c = 0\). In our exercise, the equation derived was \(6(x-14)^2 = 0\). This is already structured in a way that highlights its quadratic nature even though it's uniquely set to factor directly, reducing it to solving a simpler equation.

Understanding the basic structure of quadratic equations is crucial because they often require specific methods for solving, such as factoring, completing the square, or using the quadratic formula. In simpler cases, especially when the equation is already factored into perfect squares like \((x-14)^2\), the task becomes significantly easier, as you'll see next.

Factoring

Factoring is a method used to simplify polynomial expressions; in the context of quadratics, it involves rewriting the equation as a product of its factors. These factors are usually simpler binomial or monomial expressions.

In the original exercise, factoring was implicitly applied as the quadratic was already expressed in a factored form \((x-14)^2\). This indicates that \((x-14)\) is a repeated factor. Recognizing this pattern is key when using the zero-product property because it reveals the values that make each factor equal to zero. Understanding how to factor efficiently can turn a seemingly complex quadratic equation into a straightforward equation that you can solve with basic arithmetic.

Solving Equations

Solving equations involves finding the value of variables that make the equation true. When product factors result in a zero, the zero-product property is a powerful tool in solving equations, specifically quadratic ones. This property states that if \(a \cdot b = 0\), then either \(a = 0\) or \(b = 0\) (or both).

In the exercise provided, using this property quickly led to the conclusion. The equation \((x-14)^2 = 0\) was simplified to \(x-14 = 0\) following the zero-product principle. By solving this simpler equation, we could determine that \(x = 14\).

• This step-by-step simplification underlines the importance of understanding and applying the zero-product property in the context of quadratic equations.
• Mastering these principles provides a foundation for tackling more complex problems efficiently.