Square of a Difference Practice Test

Expand, factor, complete, and apply the identity (a−b)².

Square of a Difference Practice Test

This test has 20 questions

Sign Control Fold

The middle term turns negative. The final square does not.

The square of a difference is easy to misread because one minus sign appears in the binomial but affects the expansion in two different ways. The two cross-products are negative, so the middle term is negative and doubled. The last term remains positive because the negative second part is squared.

Sign anatomyTrack where the minus sign survives and where it disappears.

Scaled differencesSquare full coefficients and variables, not partial pieces.

Reverse factoringRecognize perfect-square trinomials with negative middle terms.

Mental shortcutsUse numbers just below convenient bases.

1. The sign pattern comes from two negative cross-products and one positive final square

The identity is structural: the minus sign appears twice in the cross-products but disappears when the second term is squared.

Core identity

{(a - b)}^{2} = a^{2} - 2 a b + b^{2}

a (- b) + (- b) a = - 2 a b

{(- b)}^{2} = b^{2}

The middle term is negative because both cross-products are negative. The last term stays positive because it is the square of the negative second part.

a^{2}

a (- b)

(- b) a

{(- b)}^{2} = b^{2}

The fold diagram separates the four multiplication pieces so the sign behavior is visible before like terms are combined.

2. A correct expansion always has positive outer squares and a negative doubled middle product

Check all three pieces before simplifying further.

First square

a^{2}

Square the complete first binomial part.

Negative middle term

- 2 a b

Twice the product appears with a negative sign.

Positive final square

b^{2}

The final sign is positive even though the second binomial part was subtracted.

3. Scaled binomials require the same sign control with complete coefficients

Treat each full binomial piece as a unit before squaring or multiplying.

Scaled sign laneSquare the first part, subtract twice the full product, then add the square of the second part.

One-variable example

{(2 x - 5)}^{2}

4 x^{2} - 20 x + 25

The coefficient on the variable is squared in the first term and also participates in the middle product.

Two-variable example

{(3 x - 2 y)}^{2}

9 x^{2} - 12 x y + 4 y^{2}

Both variables remain part of their complete binomial pieces throughout the identity.

4. Reverse recognition factors a matching negative-middle trinomial into one repeated binomial

The outer terms must be perfect squares and the middle term must be negative twice their product.

Monic perfect square

x^{2} - 14 x + 49 = {(x - 7)}^{2}

The negative middle coefficient determines the subtraction sign inside the repeated binomial.

Scaled perfect square

9 x^{2} - 24 x + 16 = {(3 x - 4)}^{2}

Include the leading coefficient inside the binomial before checking the doubled product.

5. Missing-term problems can be reconstructed from the negative-middle pattern

Once the two squared pieces are known, the middle coefficient is determined.

Reconstruction rule

For a square of a difference, the magnitude of the middle coefficient is twice the product of the binomial parts, while the sign is negative.

Missing middle coefficient

x^{2} - k x + 36

k = 2 \cdot 1 \cdot 6 = 12

Missing constant

x^{2} - 18 x + k

k = 9^{2} = 81

The sign of the middle term identifies a square of a difference rather than a square of a sum.

6. Coefficient matching must carry the negative sign through the doubled product

The magnitude comes from twice the product; the sign comes from the subtraction in the binomial.

Original square

{(4 x - 3)}^{2}

Identify the complete scaled variable part and the constant part.

middle term⇄

Signed coefficient

- 2 \cdot (4 x) \cdot 3 = - 24 x

The middle coefficient is negative because the binomial uses subtraction.

7. Numerical shortcuts are especially useful for numbers just below a round base

Rewrite the number as a convenient base minus a small adjustment.

Just below one hundred

98^{2} = {(100 - 2)}^{2}

10000 - 400 + 4 = 9604

The identity turns long multiplication into one large square, one easy subtraction, and one small square.

Just below fifty

49^{2} = {(50 - 1)}^{2}

2500 - 100 + 1 = 2401

The same pattern works around any convenient base.

8. Reverse expansion verifies the factorization in one line

Check that the repeated binomial reproduces the positive outer squares and negative middle term.

Back-check laneExpand the proposed repeated binomial mentally and compare every term.

{(x - 6)}^{2}

x^{2} - 12 x + 36

If the final square or middle sign is wrong, the factorization cannot be correct.

9. Error analysis: the last sign is the most common trap

The square of a difference is not a difference of squares.

Difference of squares substituted incorrectly

${(a - b)}^{2} \neq a^{2} - b^{2}$

Factor two omitted

The middle term comes from two equal negative cross-products.

Final square made negative

${(- b)}^{2} = b^{2}$

Coefficient not squared

Any numerical coefficient inside a binomial part is included in the square.

Reverse factoring ignores the middle sign

A negative middle term points to subtraction inside the repeated binomial.

Mental shortcut loses the positive correction

The final small square is added, not subtracted.

Final square-of-a-difference audit

Before accepting an expansion or factorization, check both outer squares and the negative doubled middle product.

Did I identify the complete first and second binomial parts?Keep coefficients and variables attached to their parts.

Did I square the first part completely?The first term is positive.

Did I subtract twice the product?The middle term must be negative and doubled.

Did I make the final square positive?Squaring removes the negative sign from the second part.

For reverse factoring, does the middle coefficient match exactly?Only then is the trinomial a perfect square of a difference.

Did I reverse-check by expanding?A quick back-check catches sign and coefficient mistakes.