Problem 15 - Beginning and Intermediate Algebra

Question

Solve using the zero-factor property. $$ x^{2}-169=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = 13\) and \(x = -13\).

Step 1 - Set the equation equal to zero

The equation is already given in the form where one side is equal to zero: \(x^{2} - 169 = 0\)

Step 2 - Factor the quadratic expression

Notice that the left-hand side is a difference of squares. Recall that a difference of squares can be factored as follows: \[a^2 - b^2 = (a - b)(a + b)\] In this case, we can rewrite \(169\) as \(13^2\) since \(13 \times 13 = 169\). So, we have: \(x^2 - 13^2 = (x - 13)(x + 13)\)

Step 3 - Set each factor equal to zero

Using the zero-factor property, if the product of two factors is zero, then at least one of the factors must be zero. Set each factor equal to zero and solve: \(x - 13 = 0\) \(x + 13 = 0\)

Step 4 - Solve for the variable

Now solve each of the equations: \(x - 13 = 0\) Adding 13 to both sides, we get: \(x = 13\) For the second equation: \(x + 13 = 0\) Subtracting 13 from both sides, we get: \(x = -13\)

Key Concepts

Quadratic Equations

Quadratic equations are equations of the form \(ax^2 + bx + c = 0\) where \(a, b,\) and \(c\) are constants with \(a \eq 0\). They are called 'quadratic' because the term 'quadratic' means 'square'.

In our exercise, we have the quadratic equation \(x^2 - 169 = 0\). Notice that there is no \(bx\) term here, which simplifies the solving process.

Quadratic equations can be solved using various methods, such as:

• Factoring
• Completing the Square
• Using the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In our example, we utilized the factoring method facilitated by recognizing the difference of squares.

Difference of Squares

The difference of squares is a specific form of a quadratic expression that fits the pattern \(a^2 - b^2 = (a - b)(a + b)\).

This pattern is very useful when solving quadratic equations, as it allows us to factor the expression into a product of two binomials.

In our exercise, we recognize that \(169\) can be written as \(13^2\), leading us to factor \(x^2 - 13^2\) as \( (x - 13)(x + 13)\).

This factoring step is crucial because it breaks down the quadratic expression into simpler components that we can then solve separately. Once factored, we can apply the zero-factor property to find the roots of the original equation.

Factoring

Factoring is a method used to break down expressions into simpler, multiplicative components.

For quadratic equations, factoring is particularly effective when dealing with simple forms like the difference of squares or when the quadratic equation easily splits into binomials.

In the given problem, \(x^2 - 13^2 = 0\) was factored into \( (x - 13)(x + 13)\), leveraging the difference of squares identification.

Once factored, we can employ the zero-factor property, which states that if the product of two factors is zero, one or both of the factors must be zero:

• Set each factor equal to zero: \( x - 13 = 0 \) and \( x + 13 = 0 \)
• Solve for \( x \) in both equations, resulting in roots \( x = 13 \) and \( x = -13 \).

This process efficiently provides the solutions to the original quadratic equation.