- Define a perfect square trinomial \(x^2 + bx + c\) by the condition \(c = (\frac{b}{2})^2\).
- Evaluate each polynomial by comparing its constant term to the square of half its middle coefficient.
- Confirm that \(x^2 - 6x + 9\) factors to \((x - 3)^2\).
- Conclude that the perfect square trinomial is \(\boxed{x^2 - 6x + 9}\).
Explanation
- Understanding Perfect Square Trinomials
A perfect square trinomial is a polynomial of the form \(ax^2 + bx + c\) that can be factored as \((dx + e)^2\). For a monic quadratic trinomial (where the coefficient of \(x^2\) is 1), the expression \(x^2 + bx + c\) is a perfect square if it can be written in the form \((x + k)^2 = x^2 + 2kx + k^2\). Comparing the coefficients, we see that for the trinomial to be a perfect square, the constant term \(c\) must be equal to the square of half the middle coefficient, which is \(\left(\frac{b}{2}\right)^2\).
- Evaluating the Polynomials
We will test each of the given polynomials to see if they satisfy the condition \(c = \left(\frac{b}{2}\right)^2\):
- For \(x^2 - 6x + 9\): Here \(b = -6\) and \(c = 9\). Calculating the condition: \(\left(\frac{-6}{2}\right)^2 = (-3)^2 = 9\). Since \(9 = 9\), this is a perfect square.
- For \(x^2 - 2x + 4\): Here \(b = -2\) and \(c = 4\). Calculating the condition: \(\left(\frac{-2}{2}\right)^2 = (-1)^2 = 1\). Since \(1
eq 4\), this is not a perfect square.
- For \(x^2 + 5x + 10\): Here \(b = 5\) and \(c = 10\). Calculating the condition: \(\left(\frac{5}{2}\right)^2 = (2.5)^2 = 6.25\). Since \(6.25
eq 10\), this is not a perfect square.
- For \(x^2 + 4x + 16\): Here \(b = 4\) and \(c = 16\). Calculating the condition: \(\left(\frac{4}{2}\right)^2 = (2)^2 = 4\). Since \(4
eq 16\), this is not a perfect square.
- Conclusion
After testing the polynomials, we found that \(x^2 - 6x + 9\) satisfies the condition to be a perfect square trinomial. Specifically, it can be factored as \((x - 3)^2\). Therefore, the polynomial that can be represented by a perfect square model is \(x^2 - 6x + 9\).
Examples
In architecture and design, visualizing area as a square is a fundamental concept. A perfect square trinomial like \(x^2 - 6x + 9\) represents the area of a square with side length \(x-3\). By understanding these patterns, engineers and architects can optimize material usage and calculate dimensions efficiently when designing square-based structures or floor plans where the area is defined by quadratic relationships.