- The expansion \((x+n)^2\) always results in \(x^2 + 2nx + n^2\), matching the form \(ax^2 + bx + c\).
- From this, we identify \(b = 2n\) and \(c = n^2\).
- Solving for \(n\), we get \(n = b/2\), which leads to the relationship \(c = (b/2)^2\).
- This confirms that \(c\) is the square of half of \(b\), which is option b.
- Final Answer: \(\boxed{\text{b}}\)
Explanation
- Analyzing the Pattern
We are given four examples of binomial expansions:
\[ \begin{array}{l} (x+3)^2 = x^2 + 6x + 9 \ (x+4)^2 = x^2 + 8x + 16 \ (x+5)^2 = x^2 + 10x + 25 \ (x+6)^2 = x^2 + 12x + 36 \end{array} \]
Each result is written in the form \(ax^2 + bx + c\), where \(a=1\). We observe the relationship between the coefficients \(b\) and \(c\) in these expressions. Specifically, we want to find how \(c\) relates to \(b\) for any expansion of \((x+n)^2 = x^2 + 2nx + n^2\).
- Identifying the Relationship between b and c
By comparing the standard form \((x+n)^2 = x^2 + 2nx + n^2\) with the general form \(ax^2 + bx + c\), we identify the coefficients:
- The coefficient of \(x^2\) is \(a = 1\).
- The coefficient of \(x\) is \(b = 2n\).
- The constant term is \(c = n^2\).
Our objective is to relate \(c\) to \(b\). Since \(b = 2n\), we can solve for \(n\) in terms of \(b\):
\[n = \frac{b}{2}\]
Substituting this expression for \(n\) into the equation for \(c\):
\[c = n^2 = \left(\frac{b}{2}\right)^2\]
This shows that \(c\) is the square of half of the coefficient \(b\).
- Verifying and Selecting the Correct Option
Now let's check the given options:
A\(c\) is the square of \(b\) (\(c = b^2\)) - Incorrect.
B\(c\) is the square of half of \(b\) (\(c = (\frac{b}{2})^2\)) - Correct.
C\(c\) is \(1.5\) times that of \(b\) (\(c = 1.5b\)) - Incorrect.
D\(c\) is double \(b\) (\(c = 2b\)) - Incorrect.
Testing with the first example, where \(n=3\):
\(b = 2(3) = 6\)
\(c = 3^2 = 9\)
Check option b: \((\frac{b}{2})^2 = (\frac{6}{2})^2 = 3^2 = 9\). This holds true.
Since the derived relationship \(c = (\frac{b}{2})^2\) holds for the identified patterns and corresponds to option b, we conclude that the correct choice is b.
Examples
In algebra, recognizing patterns in quadratic expansions is essential for tasks like completing the square or factoring expressions. For instance, if you are given the expression \(x^2 + 14x + c\) and need to make it a perfect square, you know that \(c\) must be \((\frac{14}{2})^2 = 7^2 = 49\). This technique allows you to quickly transform quadratic polynomials into vertex form, which is useful in physics for finding the maximum or minimum height of a projectile, or in economics to determine the quantity that minimizes production costs.