Algebraic identities

Correct answer: A. x² - 49

Answer: A, x² - 49. The difference-of-squares identity is a² - b² = (a - b)(a + b). Here, x² is the square of x and 49 is the square of 7, so x² - 49 = x² - 7². Applying the identity gives (x - 7)(x + 7). Option B is a sum of squares because its sign is plus, so it does not match the identity. Option C is a perfect-square trinomial: x² + 14x + 49 = (x + 7)²; it has three terms, not two squared terms separated by subtraction. Option D has terms that can be rewritten as 2(x² + 4), but it is not directly a difference of two squares. Memory cue: first check two perfect squares and then check for a minus sign between them.

Correct answer: C. a² - b² = (a - b)(a + b)

Answer: C, a² - b² = (a - b)(a + b). Rewrite 9 as 3², so x² - 9 becomes x² - 3². This has exactly the form a² - b², with a = x and b = 3. Therefore, x² - 9 = (x - 3)(x + 3). Option A is the identity for the square of a sum and would produce a middle term 2ab. Option B is the identity for the square of a difference and would produce a middle term -2ab. Option D is the distributive law for taking a common factor, not the difference-of-squares identity. A quick check is to multiply (x - 3)(x + 3): the middle terms cancel, leaving x² - 9. Memory cue: difference of squares means difference times sum.

Correct answer: A. x² - 25

Answer: A, x² - 25. The identity (a + b)(a - b) is equivalent to a² - b². Since 25 = 5², x² - 25 becomes x² - 5². Hence it factorises as (x + 5)(x - 5). Option B is x² + 5², a sum of squares, not a difference. Option C is a perfect-square trinomial because x² + 10x + 25 = (x + 5)²; it follows the square-of-a-sum identity instead. Option D has only two terms, but the second term is 5x, not a square constant, so it does not match the required pattern. The order of the two factors may be reversed without changing the product: (x + 5)(x - 5) equals (x - 5)(x + 5). Memory cue: look for two square terms separated by a minus sign, then write difference times sum.

Correct answer: A. a² + 2ab + b²

Answer: A, a² + 2ab + b². The square of the binomial a + b is (a + b)² = a² + 2ab + b². Thus the expression has the required pattern: square of the first term, twice the product of the two terms, and square of the second term. Option B does not have the required middle term 2ab; its middle term is -ab. Option C has a negative final square, so it cannot be the expansion of (a + b)² or (a - b)². Option D is a difference of two squares and factors as (a - b)(a + b), but it is not a binomial square expansion. A useful verification is to multiply (a + b) by itself and combine the two ab terms. Memory cue: a binomial square has three terms and the middle term is ±2 times the product of the two terms.

Correct answer: A. \(x^2+10x+25\)

Answer: A, \(x^2+10x+25\). Use the identity \((a+b)^2=a^2+2ab+b^2\). In the first expression, \(x^2
e a^2\) suggests a=x, and 25=5^2 suggests b=5. The middle term required is \(2ab=2(x)(5)=10x\), exactly what appears. Hence \(x^2+10x+25=(x+5)^2\), so it is the square of a binomial. Option B has the same first two terms, but the constant should be 25, not 20. Option C has the wrong middle coefficient: \((x+5)^2\) would have 10x, not 5x. Option D has no middle term and is not the displayed trinomial square; \((x+5)^2\) would again require 10x and 25. Memory cue: for \(x^2+2kx+k^2\), check whether the constant is the square of half the x-coefficient.

Correct answer: A. \(x^2+10x+25\)

Answer: A, \(x^2+10x+25\). A trinomial that is the square of a binomial follows \(a^2+2ab+b^2=(a+b)^2\). Take a=x and b=5. Then \(a^2=x^2\), \(2ab=10x\), and \(b^2=25\), so \(x^2+10x+25=(x+5)^2=(x+5)(x+5)\). The two factors are identical, as required. Option B has 20 instead of the necessary 25. Option C has 5x instead of the necessary 10x. Option D has 30 instead of 25. Thus none of B, C, or D can be the square of \(x+5\), and no other value of b can satisfy both the middle and constant terms. Memory cue: take half of the middle coefficient, square it, and compare that result with the constant term. If they match, the trinomial is a perfect square.

Correct answer: A. \(x^2-25\)

Answer: A, \(x^2-25\). The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\). Its two requirements are: both terms must be perfect squares, and they must be separated by a minus sign. Here \(x^2=x^2\) and \(25=5^2\), so \(x^2-25=x^2-5^2=(x-5)(x+5)\). Option A therefore fits exactly. Option B is a sum of squares, not a difference of squares. Option C is a perfect-square trinomial, \((x+5)^2\), so it uses a different identity. Option D is also a perfect-square trinomial, \((x-5)^2\), not a difference of two squares in the required two-term form. Memory cue: look for “square minus square”; the factors always have one minus and one plus sign.

Correct answer: A. p + q = b and pq = c

The direct answer is A: p + q = b and pq = c. Expand the proposed factors step by step: (x + p)(x + q) = x·x + x·q + p·x + p·q = x² + (p + q)x + pq. Now compare this with x² + bx + c. The coefficient of x must therefore satisfy p + q = b, and the constant term must satisfy pq = c. Option B reverses these roles, so it would give the wrong coefficient and constant term. Option C uses a difference that does not appear in the expansion and gives no correct product condition. Option D unnecessarily assumes p and q are equal; many valid factorisations use different numbers, such as (x + 3)(x + 4). Memory cue: in a monic quadratic, the pair’s sum controls the middle coefficient and its product controls the constant term.

Correct answer: A. m² − 2mn + n²

The direct answer is A: m² − 2mn + n². Start from the identity (m − n)² = (m − n)(m − n). Multiplying gives m² − mn − mn + n² = m² − 2mn + n². Thus option A has exactly the required three terms and signs. Option B has both an incorrect middle sign and an incorrect sign before n². Option C is the difference of squares, which factorises as (m − n)(m + n), not as the single square (m − n)². Option D has only mn rather than the required −2mn, so it cannot be this perfect square. Memory cue: a square of a difference always has a negative double-product middle term.

Correct answer: A. \(a^2-b^2=(a-b)(a+b)\)

Direct answer: A. The expression \(p^2-q^2\) is a difference of two perfect squares: the squares are \(p^2\) and \(q^2\), separated by a minus sign. The identity \(a^2-b^2=(a-b)(a+b)\) therefore gives \(p^2-q^2=(p-q)(p+q)\). Expanding verifies it: \((p-q)(p+q)=p^2+pq-pq-q^2=p^2-q^2\). Option A is correct. Option B is the identity for the square of a sum and creates a positive middle term. Option C is the square of a difference and contains a middle term \(-2ab\), which is absent here. Option D is for the difference of cubes, not squares, and has three-factor-degree structure. Memory cue: two squared terms with a minus sign mean difference of squares: difference bracket times sum bracket.

Correct answer: A. \(a^2-b^2\)

Direct answer: A, \(a^2-b^2\). The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\). The expression in A has two perfect-square terms, \(a^2\) and \(b^2\), joined by a minus sign, so it fits the pattern exactly. Option A therefore factors as \((a-b)(a+b)\). Option B is a three-term perfect square: \(a^2+2ab+b^2=(a+b)^2\), so it uses a different identity. Option C is a sum of squares, not a difference; the given identity cannot be applied directly. Option D is a trinomial, but its middle coefficient is 1 rather than the required 2 for the standard perfect-square pattern, and it is not the difference of two squares. Memory cue: the name itself gives the test—look for two squares separated by a minus sign.