Problem 13 - Beginning and Intermediate Algebra

Question

Solve using the zero-factor property. $$ x^{2}=121 $$

Step-by-Step Solution

Answer

The solutions are x = 11 and x = -11.

Step 1: Understand the Zero-Factor Property

The zero-factor property states that if a product of two factors is zero, then at least one of the factors must be zero. In this case, the equation needs to be brought to the form where it can be factored.

Step 2: Rewrite the Equation

Rewrite the given equation as a difference of squares: \[ x^2 - 121 = 0 \]

Step 3: Factor the Equation

Factor the difference of squares. Remember that \( a^2 - b^2 = (a - b)(a + b) \). So, \[ x^2 - 121 = (x - 11)(x + 11) = 0 \]

Step 4: Apply the Zero-Factor Property

Set each factor equal to zero: \[ x - 11 = 0 \] \[ x + 11 = 0 \]

Step 5: Solve for x

Solve each of the resulting equations: \[ x - 11 = 0 \implies x = 11 \] \[ x + 11 = 0 \implies x = -11 \]

Key Concepts

Difference of Squares

In algebra, the difference of squares is a useful technique to simplify quadratic equations. The key formula is: \[a^2 - b^2 = (a - b)(a + b)\]Here, \(a^2 - b^2\) is the difference of squares. In our example, we can identify the squared terms as \(x^2\) and \(121\), which is \(11^2\). So, we rewrite the equation as \(x^2 - 11^2 = 0\).

Factoring

Factoring is the process of breaking down an expression into simpler 'factors' that, when multiplied together, give back the original expression. Using the difference of squares formula, we factor \(x^2 - 11^2\) as \( (x - 11)(x + 11)\). Now, our original equation \(x^2 - 121 = 0\) becomes \( (x - 11)(x + 11) = 0\). This step is crucial for simplifying and solving quadratic equations.

Solving Quadratic Equations

Solving quadratic equations often involves moving terms around to apply the zero-factor property. When we have an equation in the form \( (x - 11)(x + 11) = 0\), it tells us that at least one of these factors must be zero for the product to be zero. So, we set each factor equal to zero:

• \ x - 11 = 0 \
• \ x + 11 = 0 \

From here, we isolate \(x\) to find its values.

Equation Rewriting

Equation rewriting is an important step before applying further algebraic techniques. In our problem, we started with \(x^2 = 121\). To use the difference of squares method, we rewrote it as \(x^2 - 121 = 0\). This new form allowed us to apply the factoring steps seamlessly. Rewriting equations into workable forms is often the first step in solving more complicated problems. By transforming the equation, it becomes possible to apply techniques like factoring and the zero-factor property effectively.