Quiz & Worksheet - Perfect Square Binomials | Study.com - Free Printable

Problem Analysis and Solution

The provided worksheet involves questions about perfect square binomials. Let's solve each question step by step.

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#### Question 1: \(3x^2 + 8xy + 4y^2\) is:

We need to determine whether the given expression is a perfect square binomial, just a trinomial, or something else.

##### Step 1: Understand the structure of a perfect square binomial
A perfect square binomial has the form:
\[
(a + b)^2 = a^2 + 2ab + b^2
\]
or
\[
(a - b)^2 = a^2 - 2ab + b^2
\]

For an expression to be a perfect square binomial, the coefficients must satisfy the relationship:
- The first term must be a perfect square.
- The last term must be a perfect square.
- The middle term must be twice the product of the square roots of the first and last terms.

##### Step 2: Analyze the given expression
The given expression is:
\[
3x^2 + 8xy + 4y^2
\]

- The first term is \(3x^2\).
- The last term is \(4y^2\).
- The middle term is \(8xy\).

##### Step 3: Check if it fits the perfect square binomial form
To check if this is a perfect square binomial, we need to see if:
\[
3x^2 + 8xy + 4y^2 = (ax + by)^2
\]

Expanding \((ax + by)^2\):
\[
(ax + by)^2 = a^2x^2 + 2abxy + b^2y^2
\]

Comparing coefficients:
- \(a^2 = 3\) (from \(3x^2\))
- \(b^2 = 4\) (from \(4y^2\))
- \(2ab = 8\) (from \(8xy\))

From \(b^2 = 4\), we get \(b = 2\) or \(b = -2\).

From \(a^2 = 3\), we get \(a = \sqrt{3}\) or \(a = -\sqrt{3}\).

Now, check \(2ab = 8\):
\[
2ab = 2(\sqrt{3})(2) = 4\sqrt{3} \quad \text{(not equal to 8)}
\]

Since \(2ab \neq 8\), the expression \(3x^2 + 8xy + 4y^2\) is not a perfect square binomial.

##### Step 4: Determine the correct answer
The expression is a trinomial but not a perfect square binomial. Therefore, the correct answer is:
\[
\boxed{\text{Just a trinomial}}
\]

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#### Question 2: \(81x^2 + 18xy + y^2\) is:

We need to determine whether the given expression is a perfect square binomial, just a trinomial, or something else.

##### Step 1: Analyze the given expression
The given expression is:
\[
81x^2 + 18xy + y^2
\]

- The first term is \(81x^2\).
- The last term is \(y^2\).
- The middle term is \(18xy\).

##### Step 2: Check if it fits the perfect square binomial form
To check if this is a perfect square binomial, we need to see if:
\[
81x^2 + 18xy + y^2 = (ax + by)^2
\]

Expanding \((ax + by)^2\):
\[
(ax + by)^2 = a^2x^2 + 2abxy + b^2y^2
\]

Comparing coefficients:
- \(a^2 = 81\) (from \(81x^2\))
- \(b^2 = 1\) (from \(y^2\))
- \(2ab = 18\) (from \(18xy\))

From \(a^2 = 81\), we get \(a = 9\) or \(a = -9\).

From \(b^2 = 1\), we get \(b = 1\) or \(b = -1\).

Now, check \(2ab = 18\):
\[
2ab = 2(9)(1) = 18 \quad \text{(equal to 18)}
\]

Since all conditions are satisfied, the expression \(81x^2 + 18xy + y^2\) is a perfect square binomial:
\[
81x^2 + 18xy + y^2 = (9x + y)^2
\]

##### Step 3: Determine the correct answer
The expression is both a trinomial and a perfect square binomial. Therefore, the correct answer is:
\[
\boxed{\text{A perfect square binomial}}
\]

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#### Question 3: Factor \(x^2 + 4xy + 4y^2\)

We need to factor the given expression.

##### Step 1: Analyze the given expression
The given expression is:
\[
x^2 + 4xy + 4y^2
\]

- The first term is \(x^2\).
- The last term is \(4y^2\).
- The middle term is \(4xy\).

##### Step 2: Check if it fits the perfect square binomial form
To check if this is a perfect square binomial, we need to see if:
\[
x^2 + 4xy + 4y^2 = (ax + by)^2
\]

Expanding \((ax + by)^2\):
\[
(ax + by)^2 = a^2x^2 + 2abxy + b^2y^2
\]

Comparing coefficients:
- \(a^2 = 1\) (from \(x^2\))
- \(b^2 = 4\) (from \(4y^2\))
- \(2ab = 4\) (from \(4xy\))

From \(a^2 = 1\), we get \(a = 1\) or \(a = -1\).

From \(b^2 = 4\), we get \(b = 2\) or \(b = -2\).

Now, check \(2ab = 4\):
\[
2ab = 2(1)(2) = 4 \quad \text{(equal to 4)}
\]

Since all conditions are satisfied, the expression \(x^2 + 4xy + 4y^2\) is a perfect square binomial:
\[
x^2 + 4xy + 4y^2 = (x + 2y)^2
\]

##### Step 3: Determine the correct answer
The factored form of the expression is:
\[
\boxed{(x + 2y)^2}
\]

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Final Answers:

1. \(\boxed{\text{Just a trinomial}}\)
2. \(\boxed{\text{A perfect square binomial}}\)
3. \(\boxed{(x + 2y)^2}\)