Problem 20 - Precalculus Student Solutions Manual 5th

Question

Solve each equation by the zero-factor property. $$x^{2}-64=0$$

Step-by-Step Solution

Answer

The solutions are \(x = 8\) and \(x = -8\).

Step 1: Factor the Equation

The given equation is \(x^2 - 64 = 0\). Notice that this equation is in the form of a difference of squares. Factor the left-hand side using the difference of squares formula, \(a^2 - b^2 = (a - b)(a + b)\). Here, \(a = x\) and \(b = 8\), so \(x^2 - 64 = (x - 8)(x + 8)\).

Step 2: Apply the Zero-Factor Property

The zero-factor property states that if the product of two factors is zero, then at least one of the factors must be zero. So, set each factor equal to zero: \(x - 8 = 0\) and \(x + 8 = 0\).

Step 3: Solve Each Equation

Solve the equations from the previous step: \(x - 8 = 0\) and \(x + 8 = 0\). These simplify to \(x = 8\) and \(x = -8\) respectively.

Step 4: Write the Final Solutions

The solutions to the equation \(x^2 - 64 = 0\) are \(x = 8\) and \(x = -8\).

Key Concepts

Difference of Squares

A difference of squares occurs when you have an expression of the form \(a^2 - b^2\). This can always be factored into the form \( (a - b)(a + b)\). It’s a neat property because it simplifies the solving process. In the given problem of \(x^2 - 64 = 0\), we recognize this form where \(a = x\) and \(b = 8\). Our equation becomes \( (x - 8)(x + 8) \). By recognizing and using the difference of squares, we can break down quadratic expressions easily. Identifying the difference of squares early in problems is a helpful skill in algebra.

Factoring

Factoring is breaking down an expression into simpler parts or 'factors' that, when multiplied together, give back the original expression. It is like reversing the multiplication process. For instance, in the problem \(x^2 - 64 = 0\), factoring comes in handy. We recognize it as a difference of squares and factor it as \( (x - 8)(x + 8) \). It's essential to be comfortable with various factoring techniques:

• Factoring out the greatest common factor (GCF).
• Difference of squares, as we saw in this problem.
• Trinomials and grouping for more complex expressions.

Factoring simplifies solving quadratic equations and other algebraic expressions.

Quadratic Equations

Quadratic equations are polynomial equations of the second degree, generally in the form \(ax^2 + bx + c = 0\). They can be solved by various methods:

• Factoring.
• Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \).
• Completing the square.

The given problem, \(x^2 - 64 = 0\), is a specific type called a 'pure quadratic' equation because the 'b' value is zero. We managed to solve it by factoring. Quadratic equations have significant applications in physics, engineering, and many other fields. Whether describing the trajectory of a ball, determining profit maximization, or other scenarios, understanding how to solve these equations is crucial!