Question

which are perfect square trinomials? select two options.\
\\(\square x^2 - 9\\)\
\\(\square x^2 - 100\\)\
\\(\square x^2 - 4x + 4\\)\
\\(\square x^2 + 10x + 25\\)\
\\(\square x^2 + 15x + 36\\)

Explanation:

Step1: Recall Perfect Square Trinomial Formula

A perfect square trinomial has the form \(a^2 + 2ab + b^2=(a + b)^2\) or \(a^2-2ab + b^2=(a - b)^2\), or for difference of squares (but here we focus on trinomials, so difference of squares like \(x^2 - 9=(x - 3)(x + 3)\) is a binomial, not a trinomial, so we check trinomial options: \(x^2-4x + 4\), \(x^2 + 10x + 25\), \(x^2 + 15x + 36\), \(x^2-100\) (binomial), \(x^2 - 9\) (binomial).

Step2: Check \(x^2-4x + 4\)

Compare with \(a^2-2ab + b^2\). Here, \(a = x\), \(2ab=4x\) (so \(b = 2\) since \(2\times x\times2 = 4x\)), and \(b^2 = 4\). So \(x^2-4x + 4=(x - 2)^2\), which is a perfect square trinomial.

Step3: Check \(x^2 + 10x + 25\)

Compare with \(a^2 + 2ab + b^2\). Here, \(a = x\), \(2ab = 10x\) (so \(b = 5\) since \(2\times x\times5=10x\)), and \(b^2 = 25\). So \(x^2 + 10x + 25=(x + 5)^2\), which is a perfect square trinomial.

Step4: Check \(x^2 + 15x + 36\)

Factor \(x^2 + 15x + 36\). We need two numbers that multiply to 36 and add to 15. The numbers are 12 and 3 (\(12\times3 = 36\), \(12 + 3=15\)), so \(x^2 + 15x + 36=(x + 12)(x + 3)\), not a perfect square trinomial.

Step5: Check non - trinomial options

\(x^2-100=(x - 10)(x + 10)\) (difference of squares, binomial), \(x^2 - 9=(x - 3)(x + 3)\) (difference of squares, binomial).

Answer:

The two perfect square trinomials are:

• \(x^2-4x + 4\)
• \(x^2 + 10x + 25\)