Question

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

Explanation:

Step1: Recall Perfect Square Trinomial Formula

A perfect square trinomial has the form \(a^2 \pm 2ab + b^2=(a\pm b)^2\). It has three terms, so first, eliminate binomials (two - term expressions). \(x^{2}-9=(x - 3)(x + 3)\) (difference of squares, binomial), \(x^{2}-100=(x - 10)(x + 10)\) (difference of squares, binomial), so these two are not perfect square trinomials.

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

For \(x^{2}-4x + 4\), we have \(a = x\), \(b^{2}=4\), so \(b = 2\) (since \((- 2)^2=4\) and the middle term is \(-4x=-2\times x\times2\)). Then \(x^{2}-4x + 4=(x - 2)^2\), which fits the perfect square trinomial formula.

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

For \(x^{2}+10x + 25\), \(a=x\), \(b^{2}=25\), so \(b = 5\) (since \(5^2 = 25\)) and the middle term \(10x=2\times x\times5\). Then \(x^{2}+10x + 25=(x + 5)^2\), which fits the perfect square trinomial formula.

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

If it were a perfect square trinomial, if \(a=x\), \(b^{2}=36\), then \(b = 6\) or \(b=-6\). Then \(2ab=2\times x\times6 = 12x\) or \(2ab=- 12x\), but the middle term is \(15x
eq\pm12x\), so it is not a perfect square trinomial.

Answer:

C. \(x^{2}-4x + 4\), D. \(x^{2}+10x + 25\)