Question

a perfect square trinomial can be represented by a square model with equivalent length and width. which polynomial can be represented by a perfect square model?\
\\( x^2 - 6x + 9 \\)\
\\( x^2 - 2x + 4 \\)\
\\( x^2 + 5x + 10 \\)\
\\( x^2 + 4x + 16 \\)

Explanation:

Step1: Recall perfect square formula

A perfect square trinomial has the form \((a - b)^2 = a^2 - 2ab + b^2\) or \((a + b)^2 = a^2 + 2ab + b^2\).

Step2: Analyze each option

• Option 1: \(x^2 - 6x + 9\)
• Compare with \(a^2 - 2ab + b^2\). Here, \(a = x\), \(2ab = 6x\), so \(2b = 6\) (since \(a = x\)), then \(b = 3\). And \(b^2 = 9\), which matches. So \((x - 3)^2 = x^2 - 6x + 9\).
• Option 2: \(x^2 - 2x + 4\)
• If \(a = x\), \(2ab = 2x\) implies \(b = 1\), but \(b^2 = 1

eq 4\), so not a perfect square.

• Option 3: \(x^2 + 5x + 10\)
• If \(a = x\), \(2ab = 5x\) implies \(b = \frac{5}{2}\), then \(b^2 = \frac{25}{4}

eq 10\), not a perfect square.

• Option 4: \(x^2 + 4x + 16\)
• If \(a = x\), \(2ab = 4x\) implies \(b = 2\), then \(b^2 = 4

eq 16\), not a perfect square.

Answer:

\(x^2 - 6x + 9\) (the first option)