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Chapter 9: Problem 27

\(500=k^{2}\)

Short Answer

Expert verified
\[ k = 10 \sqrt{5} \] or \[ k = -10 \sqrt{5} \]

Step by step solution

01

Set Up the Equation

Start with the given equation: \[ 500 = k^{2} \]
02

Isolate the Variable

To solve for \(k\), take the square root of both sides of the equation: \[ k = \pm \sqrt{500} \]
03

Simplify the Square Root

Simplify \( \sqrt{500} \): \[ \sqrt{500} = \sqrt{100 \times 5} = \sqrt{100} \times \sqrt{5} = 10 \sqrt{5} \]
04

Determine the Final Answer

Since taking the square root gives both positive and negative results: \[ k = 10 \sqrt{5} \text{ or } k = -10 \sqrt{5} \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Square Root Method
The square root method is a useful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). However, when there is no linear term (\(b = 0\)), the method becomes more straightforward.
For instance, in the given problem \(500 = k^2\), we can apply the square root method to isolate \(k\).
  • Start by taking the square root of both sides: \(k = \pm \sqrt{500}\).
  • Don't forget that the square root of a number always has two solutions: a positive and a negative value.
    This is essential for capturing both possible values of the variable.
    Thus, our solutions are \(k = 10 \sqrt{5}\) and \(k = -10 \sqrt{5}\).
    This method is especially helpful for equations where the squared term is isolated from the rest.
Isolating the Variable
Isolating the variable involves getting the variable alone on one side of the equation.
Let's consider the exercise \(500 = k^2\). Here鈥檚 how to isolate \(k\):
  • Notice that \(k^2\) is already separate. Thus, we can proceed with the next steps directly.
  • We need to free \(k\) by applying the inverse operation of squaring, which is taking the square root.
  • Take the square root of both sides: \(k = \pm \sqrt{500}\).
    The inverse operations help you isolate the variable quickly and provide an exact solution. Always remember to consider both positive and negative roots in such cases. This is a crucial part of solving the equation correctly.
Simplifying Radicals
Simplifying radicals is about breaking down a complex square root into its more manageable parts. Let's simplify \(\sqrt{500}\):
  • Start by recognizing any perfect squares within \(500\). Noticing that \(500 = 100 \times 5\), note that 100 is a perfect square.
  • Then break it down: \(\sqrt{500} = \sqrt{100 \times 5}\)
  • Next, separate the square root: \(\sqrt{100} \times \sqrt{5}\). \(\sqrt{100}\) is \(10\), so you get \(10\sqrt{5}\).
    This makes our original expression \(k = \pm \sqrt{500} = \pm 10\sqrt{5}\), which is greatly simplified. Breaking down radicals into identifiable parts helps in solving the equation more efficiently.
    Using these methods consistently will lead to a solid understanding of both the process and the result.

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Most popular questions from this chapter

\(x^{2}=-100\)

\(k^{2}=-72\)

Problem: Use the quadratic formula to solve \(x^{2}-3 x-7=0 .\) Incorrect Answer: \(x=\frac{-(-3) \pm \sqrt{-3^{2}-4(1)(-7)}}{2(1)}\) \(x=\frac{3 \pm \sqrt{-9+28}}{2}\) \(x=\frac{3 \pm \sqrt{19}}{2}\)

\(\sqrt{3^{2}-4(2)(-8)}\)

\(m^{2}=-4\)
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