Solving equations of the form $x^2 = k$ involves finding the values of $x$ that, when squared, equal $k$. The primary method for solving these equations is by taking the square root of both sides. Remember, when taking the square root, you must consider both the positive and negative roots, since both a positive and a negative number, when squared, will yield a positive result. Therefore, the solutions are $x = \sqrt{k}$ and $x = -\sqrt{k}$.
For example, if you have the equation $x^2 = 9$, you take the square root of both sides: $\sqrt{x^2} = \pm\sqrt{9}$. This gives you $x = 3$ and $x = -3$.
Match the terms with their definitions:
| Term | Definition |
|---|---|
| 1. Square Root | A. The opposite of squaring a number |
| 2. Variable | B. A value that, when multiplied by itself, equals a given number |
| 3. Constant | C. A symbol representing an unknown quantity |
| 4. Equation | D. A statement that two expressions are equal |
| 5. Solution | E. A fixed value that does not change |
(Answers: 1-B, 2-C, 3-E, 4-D, 5-A)
When solving the equation $x^2 = k$, we take the _______ of both sides. Remember to include both the _______ and _______ roots. This is because squaring a _______ or _______ number will result in a positive number.
(Answers: square root, positive, negative, positive, negative)
Explain why it's important to consider both positive and negative square roots when solving equations in the form $x^2 = k$. Provide an example to illustrate your explanation.
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