📚 Topic Summary
Quadratic equations in the form $(x+a)^2 = k$ are a special type of quadratic equation where a binomial, $(x+a)$, is squared and set equal to a constant, $k$. Solving these equations involves taking the square root of both sides and then isolating $x$. Remember that when taking the square root, you need to consider both the positive and negative roots! This form makes isolating x relatively straightforward.
For example, in $(x+3)^2 = 9$, we take the square root of both sides to get $x+3 = \pm 3$. This leads to two possible solutions: $x+3 = 3$ and $x+3 = -3$. Solving these gives $x = 0$ and $x = -6$.
🧠 Part A: Vocabulary
Match the term with its definition:
| Term |
Definition |
| 1. Quadratic Equation |
A. A value that, when multiplied by itself, equals a given number. |
| 2. Square Root |
B. A term of the form $(x + a)$ within parentheses. |
| 3. Constant |
C. An equation that can be written in the form $ax^2 + bx + c = 0$. |
| 4. Binomial |
D. A fixed value that does not change. |
| 5. Solution |
E. A value that makes an equation true. |
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct terms:
To solve an equation in the form $(x+a)^2 = k$, first take the _______ of both sides. Remember to consider both the _______ and _______ square roots. Then, _______ x to find the solution. The value of 'a' is a _______ term, while 'k' is the _______ term.
🤔 Part C: Critical Thinking
Explain why it's important to consider both the positive and negative square roots when solving equations in the form $(x+a)^2 = k$. What happens if you only consider one root?