Answer: Because it equals $(a - b)^2$, and any real number squared is non-negative.

Recognize the identity: $a^2 - 2ab + b^2 = (a - b)^2$. Apply the property of squares: For any real number $r$, $r^2 \geq 0$. Since $(a-b)$ is a real number, $(a-b)^2 \geq 0$. Why distractors fail: Option A is incomplete — $a^2$ and $b^2$ being positive doesn't address the subtraction of $2ab$. Option C is false when $a$ and $b$ are both positive. Option D states a consequence rather than the fundamental reason.

Answer: $12$

Apply the middle term check: For $x^2 + kx + 36$ to be a perfect square, the last term must be a perfect square: $\sqrt{36} = 6$. Then $k = 2 \times 1 \times 6 = 12$. Verify: $x^2 + 12x + 36 = (x + 6)^2$. Expanding confirms: $x^2 + 12x + 36$. ✓ Why distractors fail: Option A ($k=6$) uses $b$ instead of $2b$. Option C ($k=18$) uses $3b$. Option D ($k=36$) confuses $b^2$ with $2b$.

Answer: $k = 4$

Set up the perfect square condition: For $x^2 + kx + k$ to be a perfect square trinomial, the middle term must equal $2\sqrt{k} \cdot x$ (since the first term is $x^2$ and the last is $k$). So $k = 2\sqrt{k}$. Solve the equation: $k = 2\sqrt{k}$. Let $u = \sqrt{k}$: $u^2 = 2u \Rightarrow u^2 - 2u = 0 \Rightarrow u(u-2) = 0$. Since $k > 0$, $u = 2$, so $k = 4$. Verify: $x^2 + 4x + 4 = (x + 2)^2$ ✓. Why distractors fail: For $k=2$: middle term should be $2\sqrt{2}x \neq 2x$. For $k=6$: should be $2\sqrt{6}x \neq 6x$. For $k=8$: should be $2\sqrt{8}x = 4\sqrt{2}x \neq 8x$.

Answer: Student A is correct because the negative middle term requires a subtraction in the binomial.

Expand Student A's answer: $(6x - 5)^2 = 36x^2 - 60x + 25$ ✓. This matches. Expand Student B's answer: $(6x + 5)^2 = 36x^2 + 60x + 25$. The middle term is $+60x$, not $-60x$. ✗ Why distractors fail: Option A is wrong because $(6x-5)^2 \neq (6x+5)^2$ — squaring does not eliminate the sign difference in the cross term. Option C is wrong because $b$ must be subtracted when the middle term is negative. Option D is wrong because the expression is a valid perfect square.

Answer: The claim is valid: $A = (2t + 5)^2$, so the side length is $2t + 5$, which is linear in $t$.

Factor the expression: $4t^2 = (2t)^2$ and $25 = 5^2$. Check: $2(2t)(5) = 20t$ ✓. So $A = (2t + 5)^2$. Interpret the result: If $A = s^2$ where $s = 2t + 5$, then $s$ is a linear function of $t$, confirming the researcher's claim. Why distractors fail: Option A is wrong; a quadratic is exactly what you'd expect for a square's area with a linear side length. Option C: $(4t+5)^2 = 16t^2 + 40t + 25 \neq 4t^2 + 20t + 25$. Option D: $2(2t)(5) = 20t$, which does match the middle term, so the claim that it should be $40t$ is false.

Answer: $4p^2 - 12pq + 9q^2$

Test Option B: $4p^2 = (2p)^2$ and $9q^2 = (3q)^2$. Middle term check: $2(2p)(3q) = 12pq$. The middle term is $-12pq$ ✓. So $4p^2 - 12pq + 9q^2 = (2p - 3q)^2$. Why the others fail: Option A: $2(a)(b) = 2ab \neq ab$. Option C: $2(x)(y) = 2xy \neq 3xy$. Option D: $4n^2 = (2n)^2$ but $2(m)(2n) = 4mn \neq 5mn$.

Answer: $(x+5)^2 = x^2 + 10x + 25$, which does not equal $x^2 + 10x + 24$.

Verify the student's claim: $(x+5)^2 = x^2 + 10x + 25 \neq x^2 + 10x + 24$. The constant term is off by 1. Identify the root cause: The expression $x^2 + 10x + 24$ is not a perfect square trinomial because $24 \neq 5^2 = 25$. The middle term check fails: $2(1)(\sqrt{24})$ does not yield a whole number. Why distractors fail: Option B is circular — writing $(x+5)(x+5)$ is identical to $(x+5)^2$ and still expands to $x^2 + 10x + 25$, not $x^2 + 10x + 24$. Option C would give $x^2 - 10x + 25$, which changes both the sign of the middle term and the constant, making it even further from the original. Option D is false; $10x$ could be a valid middle term if the constant were 25.

Answer: $18$

Determine the square root of the constant: $\sqrt{81} = 9$, so $b_{\text{value}} = 9$ in the identity. Compute the middle coefficient: Using $2ab$ with $a = x$ (so coefficient 1) and $b_{\text{value}} = 9$: the middle term coefficient is $2(1)(9) = 18$. Why distractors fail: Option A uses $\sqrt{81}$ without doubling. Option C uses $3 \times 9$. Option D confuses $b^2$ with $2b$.

Answer: It is a perfect square trinomial because $2(4x)(3) = 24x$ matches the middle term.

Identify the components: $16x^2 = (4x)^2$ and $9 = 3^2$, so $a = 4x$ and $b = 3$. Verify the middle term: $2ab = 2(4x)(3) = 24x$ ✓. Therefore $16x^2 + 24x + 9 = (4x + 3)^2$. Why distractors fail: Option A describes a different expression ($16x^2 - 9$). Option C confusingly states that a zero discriminant means not factorable — a zero discriminant actually confirms a perfect square! Option D does not expand to $16x^2 + 24x + 9$.

Answer: $a^2 + 2ab + b^2 = (a + b)^2$ and $a^2 - 2ab + b^2 = (a - b)^2$

Recall the identities: The two perfect square trinomial formulas are $(a+b)^2 = a^2 + 2ab + b^2$ and $(a-b)^2 = a^2 - 2ab + b^2$. Why distractors fail: Option B swaps the signs of the identities. Option C uses $ab$ instead of $2ab$. Option D changes $+b^2$ to $-b^2$, which would not produce a perfect square.

Answer: $9x^2 - 30x + 25 = (3x - 5)^2$

Check each option against all conditions: We need $a = 9$ (binomial starts with $3x$), a negative middle term, $c < 40$, and the expansion must match. Option A verification: $(3x - 5)^2 = 9x^2 - 30x + 25$. Here $c = 25 < 40$ ✓, middle term is negative ✓, and the expansion matches ✓. Why distractors fail: Option B: $(3x - 7)^2 = 9x^2 - 42x + 49$, but $c = 49 > 40$, so it fails the constraint. Option C: $(3x - 6)^2 = 9x^2 - 36x + 36$, but the option states $9x^2 - 18x + 36$ — the middle term is wrong, so the factored form is incorrect. Option D: $(3x - 8)^2 = 9x^2 - 48x + 64$, but the option states $9x^2 - 24x + 16$ — both the middle term and constant are wrong.

Answer: $2(x + 3)^2$

Factor out the GCF: $2x^2 + 12x + 18 = 2(x^2 + 6x + 9)$. Recognize the perfect square: $x^2 + 6x + 9 = (x + 3)^2$ since $\sqrt{9} = 3$ and $2(1)(3) = 6$. So the answer is $2(x + 3)^2$. Why distractors fail: Option A introduces irrational coefficients unnecessarily. Option C: $2(x+9)(x+2) = 2(x^2 + 11x + 18) \neq 2x^2 + 12x + 18$. Option D: $(2x+6)^2 = 4x^2 + 24x + 36$, which is twice the original.

Answer: $a = 8$, $b = -3$

Determine a from the first term: $a^2x^2 = 64x^2 \Rightarrow a^2 = 64 \Rightarrow a = 8$ (taking positive root since $a$ is a coefficient). Determine b from the constant term and sign: $b^2 = 9 \Rightarrow |b| = 3$. The middle term is $2ab \cdot x = -48x$, so $2(8)(b) = -48$, giving $b = -3$. Why distractors fail: Option B has $b = 3$, which would produce $+48x$. Option C uses $a = 64$ instead of $\sqrt{64} = 8$. Option D uses $b = -9$ instead of $b = -\sqrt{9} = -3$.

Answer: $4x^2 + 12x + 9$ square metres

Set up the area expression: Area of a square $= (\text{side})^2 = (2x + 3)^2$. Expand using the pattern: $(2x + 3)^2 = (2x)^2 + 2(2x)(3) + 3^2 = 4x^2 + 12x + 9$. Why distractors fail: Option A has incorrect coefficients for every term. Option B uses $6x$ (missing the factor of 2 in $2ab$). Option D has $6$ instead of $9$ for the constant term.

Answer: $25$

Identify a and compute b: Here $a^2 = 4x^2$, so $a = 2x$. The middle term is $-20x = -2(2x)(b)$, giving $b = 5$. Find c: $c = b^2 = 5^2 = 25$. So $4x^2 - 20x + 25 = (2x - 5)^2$. Why distractors fail: Option A ($c=10$) uses $2b$ instead of $b^2$. Option B ($c=20$) confuses the middle term coefficient with $c$. Option D ($c=100$) squares the middle coefficient $10$ instead of $b = 5$.

Answer: $x^2 + 8x + 14$

Check each trinomial: For each option, verify that the first and last terms are perfect squares and the middle term equals $2ab$. Option C fails the test: $x^2 + 8x + 14$: The last term $14$ is not a perfect square, so this cannot be a perfect square trinomial. Verify the others: Option A: $(x+3)^2$ ✓. Option B: $(2x-3)^2$ ✓. Option D: $(4x+5)^2$ ✓.

Answer: $(7m - 6)^2$

Identify a and b: $49m^2 = (7m)^2$ so $a = 7m$, and $36 = 6^2$ so $b = 6$. Verify the middle term: $2ab = 2(7m)(6) = 84m$. The middle term is $-84m$, so we use $(a - b)^2 = (7m - 6)^2$. Why distractors fail: Option B has the wrong sign (middle term would be $+84m$). Option C uses $36$ instead of $\sqrt{36} = 6$. Option D uses $49m$ instead of $\sqrt{49m^2} = 7m$.

Answer: $9x^2 - 24x + 16$

Identify a and b: Here $a = 3x$ and $b = 4$. Using $(a - b)^2 = a^2 - 2ab + b^2$. Compute each term: $a^2 = (3x)^2 = 9x^2$, $2ab = 2(3x)(4) = 24x$, $b^2 = 16$. Result: $9x^2 - 24x + 16$. Why distractors fail: Option A uses $12x$ instead of $24x$, computing $2ab$ as $(3x)(4)$ without the factor of 2. Option C fails to square the coefficient 3. Option D has $-16$ instead of $+16$; the last term in a perfect square is always positive.

Answer: $x^2 + 10x + 25$

Apply the perfect square trinomial pattern: The pattern $(a + b)^2 = a^2 + 2ab + b^2$ applies here with $a = x$ and $b = 5$. Compute each term: $a^2 = x^2$, $2ab = 2(x)(5) = 10x$, $b^2 = 25$. So $(x+5)^2 = x^2 + 10x + 25$. Why distractors fail: Option A omits the middle term $2ab$. Option C uses $5x$ instead of $2(5)x = 10x$. Option D incorrectly doubles the $x^2$ term.

Answer: $25x^2 + 20xy + 4y^2$

Apply the formula: $(a + b)^2 = a^2 + 2ab + b^2$ with $a = 5x$, $b = 2y$. Compute each term: $a^2 = 25x^2$, $2ab = 2(5x)(2y) = 20xy$, $b^2 = 4y^2$. Result: $25x^2 + 20xy + 4y^2$. Why distractors fail: Option A uses $10xy$ (forgot the factor of 2). Option C fails to square $5x$ and $2y$ properly. Option D has $2y^2$ instead of $(2y)^2 = 4y^2$.

Answer: $(x - 7)^2$

Identify the pattern: The trinomial has the form $a^2 - 2ab + b^2$ with $a = x$ and $b = 7$, since $7^2 = 49$ and $2(x)(7) = 14x$. Apply the formula: $a^2 - 2ab + b^2 = (a - b)^2$, so $x^2 - 14x + 49 = (x - 7)^2$. Why distractors fail: Option B uses a plus sign, which would give $+14x$ as the middle term. Option C is a difference of squares factoring. Option D confuses $b = 7$ with $b = 49$.

Answer: $(x + y)^2$

Apply the identity: $a^2 + 2ab + b^2 = (a+b)^2$. With $a = x$ and $b = y$, we get $(x + y)^2$. Why distractors fail: Option A would give $x^2 - 2xy + y^2$. Option C is the difference of squares: $x^2 - y^2$. Option D omits the middle term entirely.

Answer: Yes, because $(x+6)^2$ and $(x-6)^2$ differ only in the sign of $b$.

Factor both expressions: $x^2 + 12x + 36 = (x + 6)^2$ and $x^2 - 12x + 36 = (x - 6)^2$. Compare the structures: Both are perfect square trinomials with $a = x$ and $b = 6$. The only difference is the sign: $(a+b)^2$ vs. $(a-b)^2$. The student's claim is valid. Why distractors fail: Options A and B are incorrect because both expressions are indeed perfect square trinomials. Option D confuses the factoring with a difference of squares: $(x+6)(x-6) = x^2 - 36$, which is entirely different.

Answer: The student is correct; $(x + 4)^2$ is a valid factoring, and the expression has a repeated root at $x = -4$.

Verify the factoring: $(x + 4)^2 = x^2 + 8x + 16$ ✓. The factoring is correct. Analyze the root: Setting $(x + 4)^2 = 0$ gives $x + 4 = 0$, so $x = -4$. This is a repeated (double) root, which is valid for quadratics. Why distractors fail: Option A: $(x+2)(x+8) = x^2 + 10x + 16 \neq x^2 + 8x + 16$. Option C is false; the expression is factorable. Option D gets the sign wrong: from $x + 4 = 0$ we get $x = -4$, not $x = 4$.

Answer: $(5y + 3)^2$

Identify squares: $25y^2 = (5y)^2$ and $9 = 3^2$. So $a = 5y$ and $b = 3$. Verify the middle term: $2ab = 2(5y)(3) = 30y$, which matches the middle term. Therefore $25y^2 + 30y + 9 = (5y + 3)^2$. Why distractors fail: Option B has a minus sign, which would give $-30y$. Option C uses $b = 9$ instead of $b = 3$. Option D fails to take the square root of 25.

Answer: Expanding $(3x + 4)^2$ gives $9x^2 + 24x + 16$, not $9x^2 + 12x + 4$.

Check the student's work: $(3x + 4)^2 = 9x^2 + 2(3x)(4) + 16 = 9x^2 + 24x + 16$. This does not match $9x^2 + 12x + 4$. Find the correct factoring: The correct factoring is $(3x + 2)^2$ since $a = 3x$, $b = 2$, $2ab = 12x$, and $b^2 = 4$. Why distractors fail: Option A is wrong because $a = 3x$ is correct ($(3x)^2 = 9x^2$). Option C is wrong because $9x^2 + 12x + 4$ is indeed a perfect square trinomial — it equals $(3x + 2)^2$. Option D is wrong because the middle term is positive, so the binomial should use a plus sign.

Answer: No, because $20$ is not a perfect square.

Check perfect square conditions: For $x^2 + bx + c$ to be a perfect square trinomial, both $x^2$ and $c$ must be perfect squares. Here $c = 20$, and $\sqrt{20}$ is not a whole number. Why the correct answer works: Since 20 is not a perfect square, the trinomial cannot fit the form $a^2 + 2ab + b^2$. Why distractors fail: Option A is wrong because being positive is necessary but not sufficient. Option B provides irrelevant factorizations. Option D is wrong because the parity of the middle term does not determine whether a trinomial is a perfect square.

Answer: It is twice the product of the square roots of the first and last terms.

Understand the structure: In $a^2 + 2ab + b^2$, the first term is $a^2$ (square root $a$) and the last term is $b^2$ (square root $b$). The middle term is $2 \times a \times b$. Why the correct answer works: $2ab$ is literally twice the product of $a$ and $b$, which are the square roots of $a^2$ and $b^2$ respectively. Why distractors fail: Option A describes $a^2 \cdot b^2 = a^2b^2$, not $2ab$. Option C describes $a^2 + b^2$, not $2ab$. Option D describes $\sqrt{a^2 \cdot b^2} = ab$, missing the factor of 2.