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Chapter 0: Problem 161

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.

Short Answer

Expert verified
The statement does not make sense because the quadratic formula is not necessary to solve this equation. A simpler way to solve this \(25 x^{2}-169=0\) is by recognizing it as a difference of two squares and applying the related formula. The solutions are \(x = \pm \frac{13}{5}\)

Step by step solution

01

Rewrite the equation

Firstly, the quadratic equation \(25x^{2}-169=0\) can be rewritten by recognizing that it represents difference of two squares. This can be written as: \((5x)^{2}-(13)^{2}=0\)
02

Apply difference of two squares formula

Next, instead of the quadratic formula, the difference of two squares formula could be applied here, which is \(a^{2}-b^{2} = (a-b)(a+b)\). Applying this formula to our equation leads to: \((5x-13)(5x+13)=0\)
03

Solve for x

Lastly, in this form, the solution for \(x\) can be determined easily by setting each expression in parentheses to zero: \(5x-13=0\) which gives \(x = 13/5\) or \(5x + 13 = 0\) which gives \(x = -13/5\). Therefore, the solution for \(x\) are \(\pm \frac{13}{5}\)

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Most popular questions from this chapter

Use the strategy for solving word problems, modeling the verbal conditions of the problem with a linear inequality. To earn an A in a course, you must have a final average of at least \(90 \% .\) On the first four examinations, you have grades of \(86 \%, 88 \%, 92 \%,\) and \(84 \% .\) If the final examination counts as two grades, what must you get on the final to earn an A in the course?

What is a rational expression?

Perform the indicated operations. Simplify the result, if possible. $$\left(\frac{2 x+3}{x+1} \cdot \frac{x^{2}+4 x-5}{2 x^{2}+x-3}\right)-\frac{2}{x+2}$$

Perform the indicated operations. $$\frac{1}{x^{n}-1}-\frac{1}{x^{n}+1}-\frac{1}{x^{2 n}-1}$$

Perform the indicated operations. Simplify the result, if possible. $$\frac{a b}{a^{2}+a b+b^{2}}+\left(\frac{a c-a d-b c+b d}{a c-a d+b c-b d} \div \frac{a^{3}-b^{3}}{a^{3}+b^{3}}\right)$$
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