Problem 166 - Algebra and Trigonometry

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 because it is a simpler process to factorize the difference of squares equation, \(25 x^{2}-169=0\), rather than using the quadratic formula. The solutions are \(x= -13/5\) and \(x= 13/5\).

Step 1: Identify the Type of Equation

The equation \(25 x^{2}-169=0\) is actually a 'difference of squares' equation. It is a quadratic equation, yes, but it's a special type that can be factored and solved more simply than using the quadratic formula.

Step 2: Factorize the Equation

Any 'difference of squares' equation takes the form of \(a^2 - b^2 = (a + b)(a - b)\). Thus, the equation \(25 x^{2}-169=0\) can be factorized into \((5x+13)(5x-13)=0\).

Step 3: Solve for the Variable

Next, set each factor to zero and solve for \(x\). Doing this gives \(5x+13=0\) (where \(x= -13/5\)) and \(5x-13=0\) (where \(x=13/5\)).