Problem 56 - Algebra 1: Concepts and Skills

Question

Solve the equation. $$ x(x-14)^{2}=0 $$

Step-by-Step Solution

Answer

The solution to the equation \(x(x - 14)^2 = 0\) are \(x = 0\) and \(x = 14\).

Step 1: Identify the Factors

In the given equation \(x(x - 14)^2 = 0\), the factors are \(x\) and \((x - 14)^2\). Both of these factors will be individually set to zero to find the values of \(x\).

Step 2: Solve for the First Factor

Setting the first factor \(x = 0\) yields \(x = 0\) as a solution.

Step 3: Solve for the Second Factor

Setting the second factor \((x - 14)^2 = 0\) equal to zero, we can simplify it by taking the square root to get \(x - 14 = 0\). From there, solve for \(x\) to get \(x = 14\).

Key Concepts

Factoring Quadratics

Factoring quadratics is a foundational technique employed to solve quadratic equations, which are equations of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a \eq 0\). To factor a quadratic equation means to rewrite it as a product of two binomials, or sometimes other polynomials, which are simpler to work with.
For example, the quadratic equation \(x(x-14)^2=0\) involves a perfect square trinomial. Factoring out the common factor—which in this case is achieved by realizing that \(x\) is one factor and \(x-14\) squared is another—simplifies the equation into identifiable parts that can be individually solved. Understanding this principle allows us to break down more complex equations into manageable sections that can then be evaluated using properties like the zero product property.

Zero Product Property

The zero product property is a critical concept when solving quadratic equations through factoring. It states that if the product of two expressions is zero, then at least one of the expressions must equal zero. This property is especially useful as it allows us to take a factored quadratic equation and split it into two or more separate equations, each of which can be solved for the variable.
For instance, with the equation \(x(x-14)^2=0\), we utilize the zero product property by setting each factor equal to zero and solving them individually—this is evident in the provided solution. The use of the zero product property is pivotal as it paves the way to find all possible solutions of a quadratic equation by providing a clear cut procedure.

Quadratic Solutions

Quadratic solutions are the values of \(x\) that satisfy the quadratic equation \(ax^2 + bx + c = 0\). These solutions can be real or complex numbers and are found by various methods, including factoring, completing the square, or using the quadratic formula. In our example, where we have \(x(x-14)^2=0\), the solutions are found by factoring and applying the zero product property.
By setting each factor equal to zero—factored from the original quadratic equation—we find that \(x=0\) and \(x=14\) are the solutions. It's essential to check each potential solution by substituting it back into the original equation to ensure it holds true. Quadratic equations can have two solutions, one solution, or no real solutions depending on the nature of the discriminant \(b^2 - 4ac\). In this case, we have a repeated solution \(x=14\) due to the squared term \(x-14\), illustrating how a quadratic equation can have a single unique solution repeated because of its multiplicity.