Problem 166 - College Algebra

Question

Determine whether each statement makes sense or does not make sense, and explain your reasoning. Because I want to solve \(25 x^{2}-169=0\) fairly quickly, I'll use the quadratic formula.

Step-by-Step Solution

Answer

The statement does not make sense, an easier method for solving this problem is to recognize it as a difference of squares.

Step 1: Identify the Type of Equation

The given problem is in the form of a quadratic equation \(ax^{2} + c =0\), where \(a\) and \(c\) are constants. The equation given is \(25x^{2} - 169 = 0\), which is a quadratic equation of the form \(ax^{2} - c = 0\).

Step 2: Recognize Type of Quadratic Equation

Here, \(a=25\) and \(c=169\). This specific form, where the equation lacks a \(x\) term, can be recognized as a difference of two squares. It can be factored into \((5x-13)(5x+13)=0\).

Step 3: Solve the Equation

For the factored equation \((5x-13)(5x+13)=0\), set each factor equal to zero and solve for \(x\), which gives \(x=2.6\) and \(x=-2.6\). This is a more straightforward method than using the quadratic formula.

Step 4: Evaluate the Initial Statement

The given statement suggested using the quadratic formula to solve the equation. While the quadratic formula would indeed arrive at the correct solution, it is more efficient in this case to recognize the equation as a difference of squares. Therefore, the statement does not make sense because a simpler method exists.