A student simplified a product of two linear binomials of the form $(ax - b)(ax + b)$ and obtained the difference of squares $49x^2 - 121$. If $a$ and $b$ are positive integers, evaluate the perfect square trinomial expansion of $(ax

A student simplified a product of two linear binomials of the form $(ax - b)(ax + b)$ and obtained the difference of squares $49x^2 - 121$ . If $a$ and $b$ are positive integers, evaluate the perfect square trinomial expansion of $(ax - b)^2$ .

Final answer

49x^2 - 140x + 121

Steps

Use the difference of squares pattern to find $a$ and $b$

The product of a sum and a difference of the same two terms follows the difference of squares identity: $(ax - b)(ax + b) = (ax)^2 - b^2 = a^2x^2 - b^2$

Matching this with the given expression $49x^2 - 121$ : $a^2x^2 - b^2 = 49x^2 - 121$

Comparing corresponding coefficients: $a^2 = 49 \implies a = 7 \quad (\text{since } a > 0)$ $b^2 = 121 \implies b = 11 \quad (\text{since } b > 0)$

Form the binomial $(ax - b)$

Substitute $a = 7$ and $b = 11$ into the binomial: $ax - b = 7x - 11$

Expand the perfect square trinomial

Use the square of a binomial formula, $(A - B)^2 = A^2 - 2AB + B^2$ , where $A = 7x$ and $B = 11$ : $(7x - 11)^2 = (7x)^2 - 2(7x)(11) + 11^2$

Compute each term:

First term: $(7x)^2 = 49x^2$
Middle term: $-2 \cdot 7x \cdot 11 = -140x$
Last term: $11^2 = 121$

Combining these terms gives: $49x^2 - 140x + 121$

Common Pitfall: A frequent error when squaring a binomial is forgetting the middle term and writing $(7x - 11)^2$ as simply $49x^2 + 121$ or $49x^2 - 121$ . Remember that squaring a binomial always produces a trinomial containing the cross-term $-2AB$ .

Educational illustration for this question

Typical errors

OverviewACEpath Teacher's Diagnosis

01 The student forgets the middle term when squaring a binomial, incorrectly expanding $(ax - b)^2$ as $a^2x^2 + b^2$ or $a^2x^2 - b^2$, arriving at $49x^2 + 121$.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
(ax - b)(ax + b) = a^2x^2 - b^2 = 49x^2 - 121 $\implies$ a^2 = 49 $\implies$ a = 7, b^2 = 121 $\implies$ b = 11
Step 2
(7x - 11)^2 = (7x)^2 + 11^2 $\implies$ 49x^2 + 121
Correct
(7x - 11)^2 = (7x)^2 - 2(7x)(11) + 11^2 = 49x^2 - 154x + 121
Squaring a binomial produces a perfect square trinomial $(A - B)^2 = A^2 - 2AB + B^2$ . You squared each term individually as $A^2 + B^2$ and missed the middle cross-term $-2(7x)(11) = -154x$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly used the difference of squares identity to determine that $a = 7$ and $b = 11$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted Middle Term in Binomial Square

You distributed the exponent of $2$ directly to both terms inside the parentheses instead of expanding $(7x - 11)(7x - 11)$ .

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

Confusing the power of a product rule $(AB)^2 = A^2B^2$ with the square of a sum or difference $(A - B)^2 = A^2 - 2AB + B^2$ , which causes the cross-term $-2AB$ to be overlooked.

REASONING PATH

Your ACEpath teacher’s suggested approach

Check binomial power structure: $(A - B)^2 = A^2 - 2AB + B^2$
Always write out all three terms: the square of the first term, twice the product of the terms, and the square of the second term.

CORRECTED PATH

ACEpath Teacher's Demonstration

Identify $A = 7x$ and $B = 11$ in the expansion formula $(A - B)^2 = A^2 - 2AB + B^2$ .
Compute the three terms: $A^2 = (7x)^2 = 49x^2$ , $-2AB = -2(7x)(11) = -154x$ , and $B^2 = 11^2 = 121$ .
Combine the terms to form the trinomial: $49x^2 - 154x + 121$ .

Final answer

49x^2 - 154x + 121

NEXT STEPS

Final Takeaways from Your Teacher

Practice expanding binomial squares by writing out $(A - B)(A - B)$ with the FOIL method whenever you are unsure.
Review the distinction between difference of squares $(A - B)(A + B) = A^2 - B^2$ and perfect square trinomials $(A - B)^2 = A^2 - 2AB + B^2$ .

OverviewACEpath Teacher's Diagnosis

02 The student forgets to multiply by 2 in the cross term of the binomial expansion, writing the middle term as $-abx$ instead of $-2abx$, resulting in $49x^2 - 77x + 121$.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
a^2 = 49 $\implies$ a = 7 \ b^2 = 121 $\implies$ b = 11
Step 2
(7x - 11)^2 = (7x)^2 - (7x)(11) + 11^2
Correct
(7x - 11)^2 = (7x)^2 - 2(7x)(11) + 11^2
When expanding a squared binomial $(A - B)^2 = A^2 - 2AB + B^2$ , the middle term must be doubled ( $-2AB$ ). You wrote $-(7x)(11)$ instead of $-2(7x)(11)$ .
Step 3
= 49x^2 - 77x + 121
Correct
= 49x^2 - 140x + 121
Your arithmetic $(7x)(11) = 77x$ was correct for the expression you wrote, but the final middle coefficient is incorrect due to the missing factor of $2$ from the previous step.

WHAT YOU DID WELL

Compliments from Your Teacher

You correctly recognized and applied the difference of squares pattern to find $a$ and $b$ .

You accurately squared the first and last terms, $(7x)^2 = 49x^2$ and $11^2 = 121$ .

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Omitted factor of 2 in binomial expansion

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You recalled the square of a difference pattern as $A^2 - AB + B^2$ rather than $A^2 - 2AB + B^2$ , omitting the doubled cross-term produced by FOIL multiplication.

REASONING PATH

Your ACEpath teacher’s suggested approach

Square of a Binomial Identity: $(A - B)^2 = A^2 - 2AB + B^2$
Always verify that the middle term includes both factors multiplied together and then doubled: $-2 \cdot A \cdot B$ .

CORRECTED PATH

ACEpath Teacher's Demonstration

(7x - 11)^2 = (7x)^2 - 2(7x)(11) + 11^2
= 49x^2 - 2(77x) + 121
= 49x^2 - 140x + 121

Final answer

49x^2 - 140x + 121

NEXT STEPS

Final Takeaways from Your Teacher

Write out the expansion using the FOIL method $(7x - 11)(7x - 11)$ whenever you want to double-check the middle term.
Practice applying $(A \pm B)^2 = A^2 \pm 2AB + B^2$ to binomials with non-unit leading coefficients.

OverviewACEpath Teacher's Diagnosis

03 The student uses $a$ and $b$ directly as 49 and 121 instead of taking their square roots, resulting in the expansion $(49x - 121)^2 = 2401x^2 - 11858x + 14641$.

Sample Student Work

Reviewed by ACEpath Teacher

Step-by-Step Review

Walk Through It with Your Teacher

Incorrect

Step 1
a = 49 , b = 121
Correct
a^2 = 49 $\implies$ a = 7 , b^2 = 121 $\implies$ b = 11
Expanding $(ax - b)(ax + b)$ yields $a^2x^2 - b^2$ . Matching this to $49x^2 - 121$ gives $a^2 = 49$ and $b^2 = 121$ . Because $a$ and $b$ are positive integers, you must take the principal square root of each number, giving $a = 7$ and $b = 11$ .
Step 2
(49x - 121)^2 = (49x)^2 - 2(49x)(121) + 121^2 $\implies$ = 2401x^2 - 11858x + 14641
Correct
(7x - 11)^2 = (7x)^2 - 2(7x)(11) + 11^2 = 49x^2 - 140x + 121
Your expansion technique and arithmetic were accurate for the values you chose, but the final expression is affected by using $a = 49$ and $b = 121$ instead of $a = 7$ and $b = 11$ .

WHAT YOU DID WELL

Compliments from Your Teacher

You recognized and applied the perfect square trinomial expansion identity $(A - B)^2 = A^2 - 2AB + B^2$ .

You accurately computed the square of the terms and the cross-product for the values you substituted.

THE ERROR

Where Your ACEpath Teacher Spotted the Mistake

Equating bases to squared coefficients

WHY IT HAPPENED

ACEpath Teacher's Diagnosis

You treated the terms in the difference of squares $49x^2 - 121$ as $a$ and $b$ rather than $a^2$ and $b^2$ , skipping the required square root step before constructing the binomial $(ax - b)$ .

REASONING PATH

Your ACEpath teacher’s suggested approach

Match Expanded Forms to Base Terms: In the difference of squares identity, $(ax - b)(ax + b) = (ax)^2 - b^2 = a^2x^2 - b^2$ , so the coefficients represent $a^2$ and $b^2$ , not $a$ and $b$ .
Always set the coefficient equal to the square of the base term, such as $a^2 = 49 \implies a = \sqrt{49} = 7$ , before building the new binomial.

CORRECTED PATH

ACEpath Teacher's Demonstration

Identify the base values from $a^2x^2 - b^2 = 49x^2 - 121$ : $a = \sqrt{49} = 7$ and $b = \sqrt{121} = 11$ .
Write the binomial expression: $(ax - b) = (7x - 11)$ .
Expand $(7x - 11)^2$ using $(A - B)^2 = A^2 - 2AB + B^2$ : $(7x)^2 - 2(7x)(11) + 11^2$ .

Final answer

49x^2 - 140x + 121

NEXT STEPS

Final Takeaways from Your Teacher

Before expanding a new binomial, pause and write out the explicit square root equation: $a = \sqrt{a^2}$ .
Compare the degrees and coefficients of $(ax)^2$ and $a^2x^2$ to keep track of which term is squared.
Practice identifying the base binomial from a given difference of squares.

Lock it in

Question

What is the expansion formula for $(A - B)^2$ ?

Answer

$A^2 - 2AB + B^2$ . A squared binomial always expands into a trinomial with a middle cross-term.

Resources

Answer and steps

Final answer

Steps

Typical errors

01 The student forgets the middle term when squaring a binomial, incorrectly expanding $(ax - b)^2$ as $a^2x^2 + b^2$ or $a^2x^2 - b^2$, arriving at $49x^2 + 121$.

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 forgets to multiply by 2 in the cross term of the binomial expansion, writing the middle term as $-abx$ instead of $-2abx$, resulting in $49x^2 - 77x + 121$.

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 uses $a$ and $b$ directly as 49 and 121 instead of taking their square roots, resulting in the expansion $(49x - 121)^2 = 2401x^2 - 11858x + 14641$.

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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