Answer: I, II, III, IV

Find each leading coefficient: (I) $a = -2$. (II) Rewritten in standard form: $\frac{3}{4}x^2 - x + 5$, so $a = \frac{3}{4} = 0.75$. (III) $a = 1$. (IV) $a = 4$. Sort from smallest to largest: $-2 < 0.75 < 1 < 4$, so the order is I, II, III, IV. Why distractors fail: Option A reverses the order. Option C swaps I and II. Option D places III before II, but $0.75 < 1$.

Answer: $5 + x + x^2$

Parse the student's claim: The claim is that three terms plus the presence of $x^2$ guarantees both (1) quadratic and (2) standard form. Show the claim partially fails: $5 + x + x^2$ has three terms and contains $x^2$, so it is quadratic. However, the terms are not in descending degree order, so it is not in standard form. This disproves the claim. Why distractors fail: Options A, C, and D are all three-term quadratics already written in descending degree order, so they do not serve as counterexamples.

Answer: They represent different expressions, so at most one can be the correct standard form of the original

Compare the two expressions: $-4x^2 + x - 7$ and $4x^2 - x + 7$ are not equivalent; every coefficient has the opposite sign. Clarify the rule about $a$: Standard form allows $a$ to be negative. The requirement is only that $a \neq 0$ and terms are in descending degree order. Both expressions are in standard form, but they are standard forms of different expressions. Why distractors fail: Option A is wrong because the two expressions are not equal. Option B is partially correct about negative $a$ but wrong to imply Student Q's is invalid — it is also in standard form, just of a different expression. Option D incorrectly states the leading coefficient must be positive.

Answer: $a = \frac{1}{2}$, $b = -1$, $c = 0$

Identify each term: The expression $\frac{1}{2}x^2 - x$ has a degree-2 term with coefficient $\frac{1}{2}$, a degree-1 term with coefficient $-1$, and no constant term. Apply the missing-term rule: Since there is no constant, $c = 0$. The expression in standard form is $\frac{1}{2}x^2 + (-1)x + 0$. Why distractors fail: Option A incorrectly assigns $c = -1$. Option B loses the negative sign on $b$. Option D swaps the coefficients between terms.

Answer: $-3x^2 + 4$

Check each condition: We need: (1) $a = -3$, (2) $b = 0$ (no linear term), (3) $c > 0$. Evaluate the correct option: $-3x^2 + 4$ has $a = -3$, $b = 0$, $c = 4 > 0$. All conditions satisfied. Why distractors fail: Option A has $b = 1 \neq 0$. Option B has $c = -7 < 0$. Option D has $a = 3$, not $-3$.

Answer: The student is correct; the $x^2$ terms cancel, giving $a = 0$, so the result is linear

Expand each part: $3(x^2 + 2) = 3x^2 + 6$ and $x(3x - 1) = 3x^2 - x$. Subtract and simplify: $3x^2 + 6 - (3x^2 - x) = 3x^2 + 6 - 3x^2 + x = x + 6$. The $x^2$ terms cancel. Assess the conclusion: Since $a = 0$ after simplification, the result is not quadratic. The student is correct that $x + 6$ is linear. Why distractors fail: Option A and D show incorrect expansions. Option C is wrong because $a = 0$ means the expression is definitionally not quadratic.

Answer: $-x^2 + 2x + 3$

Check each option for descending degree order: Standard form requires the $x^2$ term first, then the $x$ term, then the constant: $ax^2 + bx + c$. Why the correct answer works: $-x^2 + 2x + 3$ lists degree 2, then degree 1, then degree 0 — this is descending order, so it is in standard form with $a = -1$, $b = 2$, $c = 3$. Why distractors fail: Option A starts with the constant (degree 0). Option C starts with the $x$ term (degree 1). Option D places the $x$ term before the $x^2$ term.

Answer: The terms are not arranged in descending order of degree

Define standard form: Standard form requires $ax^2 + bx + c$ with terms in descending order of degree: degree 2, then degree 1, then degree 0. Why the correct answer works: In $4x + 7 - x^2$, the degree-1 term comes first and the degree-2 term is last. This violates the descending-order requirement. Why distractors fail: Option A is wrong because $a$ can be negative and the expression can still be in standard form. Option B is wrong because having a constant term is part of standard form. Option D is wrong because three terms is typical for standard form.

Answer: No, the student changed the sign of the $x$ term; it should be $3x^2 - x + 10$

Rewrite correctly: Rearranging $10 + 3x^2 - x$ in descending degree order gives $3x^2 - x + 10$. The signs must travel with their terms. Spot the error: The student wrote $+x$ instead of $-x$. When rearranging, the coefficient $b = -1$ must stay attached to the $x$ term. Why distractors fail: Option A overlooks the sign error. Option C proposes an incorrect ordering. Option D incorrectly suggests changing the constant's sign.

Answer: $a = 1$, $b = 2$, $c = -3$

Expand the product: $(x + 3)(x - 1) = x^2 - x + 3x - 3 = x^2 + 2x - 3$. Read off the coefficients: Comparing with $ax^2 + bx + c$: $a = 1$, $b = 2$, $c = -3$. Why distractors fail: Option B uses $b = -2$ (sign error when combining $-x + 3x$). Option C uses $c = 3$ (dropped the negative). Option D misidentifies $a$ as $2$.

Answer: $-9 + 3x^2$

Match to the target coefficients: We need $3x^2 + 0x + (-9) = 3x^2 - 9$ in standard form. Check Option B: $-9 + 3x^2$ rearranged in standard form is $3x^2 - 9$, giving $a = 3$, $b = 0$, $c = -9$. This matches. Why distractors fail: Option A gives $c = 9$, not $-9$. Option C is linear ($a = 0$). Option D rearranges to $-3x^2 + 9$, giving $a = -3$ and $c = 9$.

Answer: $3$

Recall the definition of the leading coefficient: In standard form $ax^2 + bx + c$, the leading coefficient is the value of $a$, which multiplies the $x^2$ term. Why the correct answer works: In $3x^2 + 7x - 5$, the coefficient of $x^2$ is $3$, so $a = 3$. Why distractors fail: Option A ($7$) is the value of $b$, not $a$. Option C ($-5$) is the constant term $c$. Option D ($x^2$) is the variable part, not the coefficient.

Answer: $k = 0$

Recall the requirement for $a$: In $ax^2 + bx + c$, the expression is quadratic only if $a \neq 0$. Apply to the parameter $k$: If $k = 0$, the $x^2$ term vanishes and the expression becomes $5x - 2$, which is linear. Any other nonzero $k$ keeps the expression quadratic. Why distractors fail: Options A, C, and D all give nonzero values of $k$, which preserve the $x^2$ term and keep the expression quadratic.

Answer: The student separated the negative sign from the coefficient; the correct value is $a = -1$

Identify the misconception: The negative sign in $-x^2$ is part of the coefficient. Writing $-x^2$ is the same as $(-1)x^2$, so $a = -1$. Why the correct answer works: The student incorrectly treated the sign as separate. Coefficients always include their sign, so $a = -1$. Why distractors fail: Option A is true but does not describe the student's error. Option C is irrelevant to the mistake about $a$. Option D is wrong because negative leading coefficients are perfectly valid.

Answer: $a = -2$, $b = 9$, $c = 4$

Apply the standard form template: Standard form is $ax^2 + bx + c$. Match each term: the $x^2$ coefficient is $a$, the $x$ coefficient is $b$, and the constant is $c$. Why the correct answer works: In $-2x^2 + 9x + 4$: $a = -2$, $b = 9$, $c = 4$. The negative sign belongs to $a$. Why distractors fail: Option A drops the negative sign from $a$. Option C swaps $a$ and $b$. Option D swaps $b$ and $c$.

Answer: 2

Evaluate each expression: (I) $0x^2 + 4x - 1$: since $a = 0$, this is linear, not quadratic. (II) $x^2$: $a = 1 \neq 0$, quadratic. (III) $-x^2 + x$: $a = -1 \neq 0$, quadratic. (IV) $3x + 7$: no $x^2$ term, linear. Count the quadratics: Expressions II and III are quadratic, giving a count of 2. Why distractors fail: Option A (1) likely counts only II, missing that negative $a$ is valid. Option C (3) likely counts I, ignoring that $a = 0$ disqualifies it. Option D (4) counts all, including the linear expressions.

Answer: $2x^2 - 3x + 6$; $a = 2$, $b = -3$, $c = 6$

Rearrange into descending degree order: Move the $x^2$ term first, then the $x$ term, then the constant: $6 - 3x + 2x^2 = 2x^2 - 3x + 6$. Read off the coefficients: Comparing $2x^2 - 3x + 6$ with $ax^2 + bx + c$: $a = 2$, $b = -3$, $c = 6$. Why distractors fail: Option B loses the negative sign on $b$. Option C is not in standard form because the $x$ term precedes the $x^2$ term. Option D reads the coefficients by position in the original order rather than by degree.

Answer: Because $a = 0$ would eliminate the $x^2$ term, reducing the degree below 2

Understand the role of the leading coefficient: A quadratic expression must have degree 2. The $x^2$ term is the only term that provides degree 2, and its presence depends on $a \neq 0$. Why the correct answer works: If $a = 0$, the expression becomes $bx + c$, which is linear (degree 1) or constant (degree 0), not quadratic. Why distractors fail: Option A is wrong because $c$ is unaffected by $a$. Option C confuses expressions with equations and solution counts. Option D refers to the quadratic formula, which applies to equations, not expressions.

Answer: No, both have $a = -5$, $b = 20$, $c = 0$; writing $+0$ is optional

Understand implicit zero terms: In $-5t^2 + 20t$, the constant term $c = 0$ is implicitly present. Writing $+0$ explicitly does not change the expression. Why the correct answer works: Both forms represent the same expression with $a = -5$, $b = 20$, $c = 0$. The absence of a written constant simply means $c = 0$. Why distractors fail: Option A incorrectly says $c$ becomes undefined; $c = 0$ is defined. Option B claims different standard forms, but they are identical. Option D incorrectly says the expression loses its quadratic status; $c = 0$ does not affect the degree.

Answer: $b = 0$, $c = -12$

Recognize a missing term: When a term is absent, its coefficient is zero. Here the $x$ term is missing, so $b = 0$. Identify c correctly: The constant term is $-12$, so $c = -12$. The subtraction sign belongs to the value of $c$. Why distractors fail: Option A assigns $-12$ to $b$ instead of $c$. Option B confuses $a$ with $b$. Option D drops the negative sign from $c$.

Answer: The term with no variable attached

Define each part of standard form: In $ax^2 + bx + c$, $a$ is the $x^2$ coefficient, $b$ is the $x$ coefficient, and $c$ is the constant — the term that has no variable. Why the correct answer works: $c$ is the standalone number that does not multiply any variable, making it the constant term. Why distractors fail: Option A describes $a$. Option B describes $b$. Option D confuses $c$ with the degree, which is always 2 for a quadratic.

Answer: $2x^2 - 8x + 3$; $a = 2$, $b = -8$, $c = 3$

Expand the expression: $2x(x - 4) + 3 = 2x^2 - 8x + 3$. Read off coefficients: In standard form $ax^2 + bx + c$: $a = 2$, $b = -8$, $c = 3$. Why distractors fail: Option A gives $b = -4$, only multiplying $2 \cdot 1$ instead of $2 \cdot (-4) = -8$. Option C has the wrong sign for $b$. Option D changes the sign of $c$.

Answer: $7x^2$

Determine what makes an expression quadratic: An expression is quadratic if it can be written as $ax^2 + bx + c$ with $a \neq 0$. Missing $b$ or $c$ terms are allowed. Why the correct answer works: $7x^2$ has $a = 7$, $b = 0$, $c = 0$. Since $a \neq 0$, it is quadratic even though both $b$ and $c$ are zero. Why distractors fail: Option A ($5x + 3$) is linear (degree 1). Option C ($-4x$) is also linear. Option D ($12$) is a constant (degree 0). None of these contain an $x^2$ term.