Question

which of the following expressions are perfect-square trinomials? check all of the boxes that apply.

(x^2 - 16x - 64)
(4x^2 + 12x + 9)
(x^2 + 20x + 100)
(x^2 + 4x + 16)

Explanation:

🆕 New Concept Discovered: Perfect-Square Trinomials
An expression that factors into a single squared binomial.

Step 1: Understand the Perfect-Square Trinomial Form

A trinomial is a perfect square if it can be written in the form:

$$ a^2 + 2ab + b^2 = (a + b)^2 $$

or

$$ a^2 - 2ab + b^2 = (a - b)^2 $$

To check if a trinomial \( Ax^2 + Bx + C \) is a perfect square:

1. The first term \( Ax^2 \) and the last term \( C \) must be perfect squares (so \( A > 0 \) and \( C > 0 \)). Let \( a = \sqrt{A}x \) and \( b = \sqrt{C} \).
2. The middle term must be equal to \( \pm 2ab \).

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Step 2: Test the First Expression

Expression: \( x^2 - 16x - 64 \)

• The first term is \( x^2 \), which is a perfect square: \( a = x \).
• The last term is \( -64 \). Since it is negative, it cannot be the square of a real number.

Thus, \( x^2 - 16x - 64 \) is not a perfect-square trinomial.

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Step 3: Test the Second Expression

Expression: \( 4x^2 + 12x + 9 \)

• The first term is \( 4x^2 \), which is a perfect square: \( a = 2x \).
• The last term is \( 9 \), which is a perfect square: \( b = 3 \).
• Check the middle term:

$$ 2ab = 2(2x)(3) = 12x $$

Since the middle term matches \( 12x \), this is a perfect-square trinomial:

$$ 4x^2 + 12x + 9 = (2x + 3)^2 $$

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Step 4: Test the Third Expression

Expression: \( x^2 + 20x + 100 \)

• The first term is \( x^2 \), which is a perfect square: \( a = x \).
• The last term is \( 100 \), which is a perfect square: \( b = 10 \).
• Check the middle term:

$$ 2ab = 2(x)(10) = 20x $$

Since the middle term matches \( 20x \), this is a perfect-square trinomial:

$$ x^2 + 20x + 100 = (x + 10)^2 $$

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Step 5: Test the Fourth Expression

Expression: \( x^2 + 4x + 16 \)

• The first term is \( x^2 \), which is a perfect square: \( a = x \).
• The last term is \( 16 \), which is a perfect square: \( b = 4 \).
• Check the middle term:

$$ 2ab = 2(x)(4) = 8x $$

Since the middle term is \( 4x \) and not \( 8x \), this is not a perfect-square trinomial.

Answer:

The perfect-square trinomials are:

• \( 4x^2 + 12x + 9 \)
• \( x^2 + 20x + 100 \)