A student incorrectly claims that $(a - b)^2 = a^2 - b^2$. To test this misconception, let $a = 2x + 3$ and $b = 2x - 3$. Calculate the exact algebraic difference between the student's incorrect formula $(a^2 - b^2)$ and the correct expanded formula $(a - b)^2$, expressing the result in terms of $x$.

A student incorrectly claims that $(a - b)^2 = a^2 - b^2$ . To test this misconception, let $a = 2x + 3$ and $b = 2x - 3$ . Calculate the exact algebraic difference between the student's incorrect formula $(a^2 - b^2)$ and the correct expanded formula $(a - b)^2$ , expressing the result in terms of $x$ .

Final answer

24x - 36

Steps

Find the value of $a - b$ and $a + b$

To find the algebraic difference between the incorrect expression $(a^2 - b^2)$ and the correct expanded expression $(a - b)^2$ , we can simplify each expression first using special product formulas.

Given: $a = 2x + 3$ $b = 2x - 3$

Calculate the sum and difference of $a$ and $b$ : $a - b = (2x + 3) - (2x - 3) = 2x + 3 - 2x + 3 = 6$ $a + b = (2x + 3) + (2x - 3) = 4x$

Evaluate the incorrect formula $(a^2 - b^2)$

Using the difference of squares pattern, $a^2 - b^2 = (a - b)(a + b)$ : $a^2 - b^2 = (6)(4x) = 24x$

Evaluate the correct formula $(a - b)^2$

Using the square of a binomial pattern, or directly substituting the evaluated difference $(a - b) = 6$ : $(a - b)^2 = 6^2 = 36$

(Note: Expanding algebraically yields $a^2 - 2ab + b^2$ , which also simplifies to $36$ . A common pitfall is the "freshman's dream" error, where students distribute the exponent across subtraction and drop the middle term $-2ab$ .)

Calculate the difference between the two expressions

Subtract the correct formula from the incorrect formula: $\text{Difference} = (a^2 - b^2) - (a - b)^2$ $\text{Difference} = 24x - 36$

Alternatively, by direct algebraic expansion: $(a^2 - b^2) - (a - b)^2 = (a^2 - b^2) - (a^2 - 2ab + b^2) = 2ab - 2b^2 = 2b(a - b)$ Substituting $b = 2x - 3$ and $(a - b) = 6$ : $2(2x - 3)(6) = 12(2x - 3) = 24x - 36$

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student subtracts in reverse order, calculating the correct formula minus the incorrect formula instead of the incorrect minus the correct.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Partially correct

Step 1
a^2 - b^2 = (2x + 3)^2 - (2x - 3)^2 = 24x
Step 2
a - b = (2x + 3) - (2x - 3) = 6 $\implies$ (a - b)^2 = 6^2 = 36
Step 3
$\text{Difference}$ = (a - b)^2 - (a^2 - b^2)
Correct
$\text{Difference}$ = (a^2 - b^2) - (a - b)^2
The problem asks for the difference between the incorrect formula $(a^2 - b^2)$ and the correct formula $(a - b)^2$ , which translates mathematically to $(a^2 - b^2) - (a - b)^2$ , not $(a - b)^2 - (a^2 - b^2)$ .
Step 4
$\text{Difference} = 36 - 24x$
Correct
$\text{Difference} = 24x - 36$
Due to setting up the subtraction in reverse order in the previous step, your final evaluated difference has the opposite sign.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly substituted the linear expressions for $a$ and $b$ into both formulas.

You skillfully simplified the difference of squares $(2x+3)^2 - (2x-3)^2$ to obtain $24x$ .

You correctly determined that $a - b = 6$ and squared it to find $36$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Reversed Subtraction Order

You set up the difference as $(\text{correct}) - (\text{incorrect})$ instead of $(\text{incorrect}) - (\text{correct})$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

In mathematical English, 'the difference between $A$ and $B$ ' strictly specifies the order $A - B$ . Translating this phrase into subtraction requires preserving the exact order of the two named quantities.

REASONING PATH

Your ACEpath teacher’s suggested approach

Translate Order-Sensitive Phrases Carefully: The verbal phrase 'the difference between $A$ and $B$ ' translates algebraically to $A - B$ .
Circle the first quantity as $A$ and the second quantity as $B$ , then write the subtraction skeleton $(A) - (B)$ before substituting their values.

CORRECTED PATH

ACEpath Teacher's Demonstration

Identify the minuend and subtrahend from the question text: the first quantity is the incorrect formula $(a^2 - b^2)$ , and the second quantity is the correct formula $(a - b)^2$ .
Write the difference expression in the specified order: $\text{Difference} = (a^2 - b^2) - (a - b)^2$ .
Substitute the simplified expressions into the formula: $\text{Difference} = 24x - 36$ .

Final answer

24x - 36

NEXT STEPS

Final Takeaways from Your Teacher

Review key English subtraction phrases such as 'difference between $A$ and $B$ ' ( $A - B$ ) versus 'subtract $A$ from $B$ ' ( $B - A$ ).
Practice highlighting the two terms in word problems to avoid sign reversals in non-commutative operations.

OverviewACEpath Teacher's Diagnosis

02 The student fails to distribute the negative sign when expanding (2x + 3)² - (2x - 3)², leading to an incorrect result for a² - b².

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
a^2 = (2x + 3)^2 = 4x^2 + 12x + 9 $\implies$ b^2 = (2x - 3)^2 = 4x^2 - 12x + 9
Step 2
a^2 - b^2 = 4x^2 + 12x + 9 - 4x^2 - 12x + 9 = 18
Correct
a^2 - b^2 = (4x^2 + 12x + 9) - (4x^2 - 12x + 9) = 24x
When subtracting $b^2$ , you did not enclose $4x^2 - 12x + 9$ in parentheses. Consequently, the negative sign was applied only to the first term rather than distributed to $-12x$ and $+9$ .
Step 3
(a - b)^2 = (2x + 3 - 2x + 3)^2 = 6^2 = 36
Step 4
$\text{Difference}$ = 18 - 36 = -18
Correct
$\text{Difference} = 24x - 36$
Your final subtraction structure was set up properly, but it inherited the incorrect value $18$ from Step 2 instead of $24x$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded both binomial squares $(2x + 3)^2$ and $(2x - 3)^2$ completely and accurately.

You correctly simplified $a - b$ by distributing the negative sign across $(2x - 3)$ to arrive at $6$ .

You correctly squared the resulting constant to evaluate $(a - b)^2 = 36$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Sign distribution error when subtracting polynomials

You forgot to keep parentheses around $b^2$ when writing $a^2 - b^2$ , which left the second and third terms with unchanged signs.

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Failing to treat a subtracted multi-term polynomial as a single grouped quantity requiring the distributive property: $-(p + q + r) = -p - q - r$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Enclose Subtracted Polynomials: Whenever you subtract an expression with two or more terms, always place parentheses around the entire expression: $-(A - B + C) = -A + B - C$ .
Write $-(4x^2 - 12x + 9)$ explicitly before combining terms.
Use Special Product Factorization as a Check: The difference of squares factors as $a^2 - b^2 = (a - b)(a + b)$ .
Notice that $a - b = 6$ and $a + b = 4x$ , giving $a^2 - b^2 = (6)(4x) = 24x$ directly without full polynomial expansion.

CORRECTED PATH

ACEpath Teacher's Demonstration

Substitute $a^2$ and $b^2$ with grouping parentheses: $a^2 - b^2 = (4x^2 + 12x + 9) - (4x^2 - 12x + 9)$ .
Distribute the negative sign across each term inside the second pair of parentheses: $4x^2 + 12x + 9 - 4x^2 + 12x - 9$ .
Combine like terms: $(4x^2 - 4x^2) + (12x + 12x) + (9 - 9) = 24x$ .
Subtract $(a - b)^2 = 36$ from $a^2 - b^2 = 24x$ : $\text{Difference} = 24x - 36$ .

Final answer

24x - 36

NEXT STEPS

Final Takeaways from Your Teacher

Form the habit of writing grouping parentheses around any multi-term expression immediately after a minus sign.
Practice distributing negative signs across negative terms (such as $-(-12x) = +12x$ ) systematically before combining like terms.
Compare direct expansion with factoring patterns like $(a - b)(a + b)$ to verify your algebraic calculations quickly.

OverviewACEpath Teacher's Diagnosis

03 The student forgets to square the difference when evaluating (a - b)², using a - b = 6 instead of 36.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
a^2 - b^2 = (4x^2 + 12x + 9) - (4x^2 - 12x + 9) $\implies$ = 24x
Step 2
a - b = 2x + 3 - (2x - 3) = 6
Step 3
(a - b)^2 = 6
Correct
(a - b)^2 = 6^2 = 36
You set $(a - b)^2$ equal to the value of $a - b$ itself ( $6$ ) rather than squaring it to obtain $6^2 = 36$ .
Step 4
$\text{Difference} = 24x - 6$
Correct
$\text{Difference} = 24x - 36$
Your subtraction step followed logically from your value in Step 3, but inherited the arithmetic mistake.

WHAT YOU DID WELL

Compliments from Your Teacher

You expanded both quadratic binomials accurately to get $a^2 - b^2 = 24x$ .

You evaluated the difference of linear expressions $a - b$ correctly to get $6$ .

You set up the final difference expression using your evaluated values.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted Exponent Operation

You wrote $(a - b)^2 = 6$ by directly copying the value of $a - b$ instead of computing $6^2$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

When substituting a pre-evaluated sub-expression into an outer power, you omitted the exponentiation step on the evaluated numeric value.

REASONING PATH

Your ACEpath teacher’s suggested approach

Substitute Inside the Power: If $u = a - b$ , then $u^2 = (a - b)^2$ . Always write $(u)^2$ before dropping parentheses.
Write $(6)^2$ explicitly on your paper before evaluating it as $36$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

Substitute $a - b = 6$ into $(a - b)^2$ : $(6)^2 = 36$ .
Subtract the correct square from the difference of squares: $(a^2 - b^2) - (a - b)^2 = 24x - 36$ .

Final answer

24x - 36

NEXT STEPS

Final Takeaways from Your Teacher

Always write out parentheses with the exponent when substituting a simplified value into a power, such as $(6)^2$ .
Verify edge values by expanding algebraically first: $(a^2 - b^2) - (a - b)^2 = 2ab - 2b^2$ .

Lock it in

Question

How do you translate the mathematical phrase 'the difference between $A$ and $B$ ' into an algebraic expression?

Answer

It translates strictly to $A - B$ .

Subtraction is non-commutative, so the quantity mentioned first is always the minuend.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student subtracts in reverse order, calculating the correct formula minus the incorrect formula instead of the incorrect minus the correct.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

02 The student fails to distribute the negative sign when expanding (2x + 3)² - (2x - 3)², leading to an incorrect result for a² - b².

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

03 The student forgets to square the difference when evaluating (a - b)², using a - b = 6 instead of 36.

Sample Student Work

Step-by-Step Review

WHAT YOU DID WELL

THE ERROR

WHY IT HAPPENED

REASONING PATH

CORRECTED PATH

NEXT STEPS

Lock it in

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