Problem 23 - Algebra 1: Concepts and Skills

Question

Use the zero-product property to solve the equation. \((w-17)^{2}=0\)

Step-by-Step Solution

Answer

The solution to the equation is \(w = 17\).

Step 1: Writing the equation

The given equation is \((w-17)^{2}=0\).

Step 2: Applying the zero-product property

According to the zero-product property, if a product of factors is zero, one or more of the factors must be zero. So, set \(w-17 = 0\).

Step 3: Solving for 'w'

Adding 17 to both sides of the equation gives the value of 'w' as \(w = 17\).

Key Concepts

Understanding Quadratic Equations

Quadratic equations are a fundamental part of algebra, representing equations that involve the square of an unknown variable. The standard form of a quadratic equation is given by:\[ ax^2 + bx + c = 0 \]Here, \( a \), \( b \), and \( c \) are constants, and \( x \) represents the unknown variable. Quadratic equations curve into a U-shape or parabola when graphed, where \( a \) determines the direction of the parabola.

• If \( a \) is positive, the parabola opens upwards.
• If \( a \) is negative, the parabola opens downwards.

Solving quadratic equations involves finding the value(s) of \( x \) that make the equation true. In the exercise, the equation \((w-17)^2 = 0\) can be seen as a special case of a quadratic equation, where the squared term can be expanded if needed, resembling the basic structure of quadratics.

Solving Equations Using the Zero-Product Property

Solving equations is all about finding the value(s) of the variable that satisfy the equation. The zero-product property is a powerful tool in algebra for solving equations that involve factors. It states:*If the product of two or more factors equals zero, at least one of the factors must be zero.*
This property is particularly useful in breaking down complex equations into solvable parts. In the exercise, the equation \((w-17)^2 = 0\) indicates that the product \( (w-17)(w-17) = 0 \). By applying the zero-product property:

• Each factor \( (w-17) \) can be set to zero separately.
• This simplifies solving the equation to finding when \( w-17 = 0 \).

This straightforward approach makes it easier to solve the equation and find the value of \( w \).

Factorization

Factorization is the process of breaking down an expression into its simplest components or factors. This is a key step in solving many algebraic equations and is essential for applying the zero-product property effectively. In cases of quadratic equations like \((w-17)^2 = 0\), the factorized form is already evident as \((w-17)(w-17)\). This expression indicates that the same factor repeats.Factorization can take several forms depending on the expression:

• Factor out common terms.
• Use techniques like factoring by grouping or recognizing special patterns (e.g., difference of squares).
• Apply the quadratic formula if necessary.

Mastering these techniques simplifies the process of solving equations and allows you to easily apply properties like the zero-product property.