Question

which products result in a difference of squares or a perfect square trinomial? check all that apply
(5x + 3)(5x - 3)
(7x + 4)(7x + 4)
(2x + 1)(x + 2)
(4x - 6)(x + 8)
(x - 9)(x - 9)
(-3x - 6)(-3x + 6)

Explanation:

Step1: Recall Formulas

• Difference of squares: \((a + b)(a - b)=a^2 - b^2\)
• Perfect square trinomial: \((a + b)^2=a^2 + 2ab + b^2\), \((a - b)^2=a^2 - 2ab + b^2\)

Step2: Analyze \((5x + 3)(5x - 3)\)

Using difference of squares formula (\(a = 5x\), \(b = 3\)):
\((5x+3)(5x - 3)=(5x)^2-3^2 = 25x^2 - 9\) (Difference of squares)

Step3: Analyze \((7x + 4)(7x + 4)\)

This is \((7x + 4)^2\). Using perfect square trinomial formula (\(a = 7x\), \(b = 4\)):
\((7x + 4)^2=(7x)^2+2\times7x\times4 + 4^2=49x^2+56x + 16\) (Perfect square trinomial)

Step4: Analyze \((2x + 1)(x + 2)\)

Multiply out: \(2x\times x+2x\times2+1\times x + 1\times2=2x^2+4x+x + 2=2x^2+5x + 2\) (Not difference of squares or perfect square trinomial)

Step5: Analyze \((4x - 6)(x + 8)\)

Multiply out: \(4x\times x+4x\times8-6\times x-6\times8=4x^2+32x-6x - 48=4x^2+26x - 48\) (Not difference of squares or perfect square trinomial)

Step6: Analyze \((x - 9)(x - 9)\)

This is \((x - 9)^2\). Using perfect square trinomial formula (\(a = x\), \(b = 9\)):
\((x - 9)^2=x^2-2\times x\times9 + 9^2=x^2-18x + 81\) (Perfect square trinomial)

Step7: Analyze \((-3x - 6)(-3x + 6)\)

Using difference of squares formula (\(a=-3x\), \(b = 6\)):
\((-3x-6)(-3x + 6)=(-3x)^2-6^2 = 9x^2 - 36\) (Difference of squares)

Answer:

A. \((5x + 3)(5x - 3)\)
B. \((7x + 4)(7x + 4)\)
E. \((x - 9)(x - 9)\)
F. \((-3x - 6)(-3x + 6)\)