Algebraic identities

Correct answer: B. (x^2+4x-12)

The product of two binomials can be expanded by multiplying every term in the first bracket by every term in the second bracket. In \\((x+6)(x-2)\\), the products are \\(x\cdot x=x^2\\), \\(x\cdot(-2)=-2x\\), \\(6\cdot x=6x\\), and \\(6\cdot(-2)=-12\\). Combining the middle terms gives \\(-2x+6x=4x\\).

Therefore, the complete expansion is \\(x^2+4x-12\\), so option B is correct. A useful check is the pattern \\((x+r)(x+s)=x^2+(r+s)x+rs\\). Here, \\(r+s=6-2=4\\) and \\(rs=6(-2)=-12\\). Option A incorrectly adds 6 and 2, ignoring the negative sign.

Correct answer: C. (1600)

The expression is a difference of two squares, so use \\(a^2-b^2=(a+b)(a-b)\\). Taking \\(a=104\\) and \\(b=96\\), we get \\(104^2-96^2=(104+96)(104-96)=200\\times8=1600\\). This is faster and safer than calculating both large squares separately.

Therefore, option C is correct. The key is to preserve the subtraction and use the sum and difference of the two numbers. The other values do not result from this product: \\(800\\) is only the difference multiplied by 100, \\(1200\\) is too small, and \\(2000\\) incorrectly uses the sum without the correct difference. The identity directly confirms 1600.

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

The expansion of \((a+b)^2\) contains the middle term \(+2ab\), twice the product of \(a\) and \(b\). In \((a-b)^2\), the middle term is \(-2ab\). Exam tip: the sign between the terms determines the sign of the middle term.

Correct answer: B. From (2\times a\times3b)

The direct answer is B: the term 6ab comes from the pair product 2 × a × 3b. For three terms, use the identity \\(p+q+r\\)^2=p^2+q^2+r^2+2pq+2qr+2rp\\). Here the three terms are a, 3b, and c. The pair a and 3b produces 2(a)(3b)=6ab. Thus the required mixed term is formed by doubling the product of those two terms. Option A, a², is only the square of the first term and has no b. Option B is correct because 2 × a × 3b simplifies to 6ab. Option C gives 2(3b)c=6bc, not 6ab, because it uses the second and third terms. Option D gives (3b)²=9b², which is a square term, not the mixed term involving a and b. The same expansion also contains a², 9b², c², 6ab, 6bc, and 2ac. Memory cue: in a square of several terms, every pair contributes twice its product. Decision: pass; the algebra and supplied option B are correct.

Correct answer: B. (12x)

(2\times2x\times3=12x) gives the linear (x) term. Exam tip: identify the mixed term formed with the constant separately.

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

\(x^2-25=x^2-5^2\), so it is a difference of two squares and factorises as \((x-5)(x+5)\). Option D is a perfect-square trinomial. Exam tip: check for two squared terms separated by a minus sign.

Correct answer: A. \(x^2-y^2=(x-y)(x+y)\)

The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\), so option A is correct. Option B gives \((x-y)^2=x^2-2xy+y^2\), not \(x^2-y^2\). Exam tip: use conjugate factors for a difference of squares.

Correct answer: A. (8091)

(87\times93=(90-3)(90+3)=90^2-3^2=8091). Exam tip: use (a^2-b^2) for numbers around the same centre.

Correct answer: B. (16x^2)

The direct answer is option B: \(16x^2\). This is an opposite-sign pair, so use \((a-b)(a+b)=a^2-b^2\). Taking \(a=4x\) and \(b=3\), \((4x-3)(4x+3)=(4x)^2-3^2=16x^2-9\). The extra \(+9\) cancels the \(-9\): \(16x^2-9+9=16x^2\). Option A stops before adding the final 9, so it is incomplete. Option B includes the cancellation and is correct. Option C adds 9 instead of cancelling it. Option D changes a quadratic expression into an unrelated linear expression, so it cannot be right. Exam cue: whenever two brackets have the same first terms and opposite signs, check difference of squares first.

Correct answer: A. (14) and (9)

To find both requested values, first expand the product or use the pattern for two binomials. The constant term comes from multiplying the two constants, 2 and 7, giving 14. The coefficient of x comes from the two middle products: x multiplied by 7 and 2 multiplied by x. These give 7x and 2x, whose sum is 9x. Therefore, the constant term and coefficient of x are 14 and 9 respectively, so option A is correct.

In full form, \\( (x+2)(x+7)=x^2+7x+2x+14=x^2+9x+14 \\). The number at the end is the constant term, while the number attached to x is its coefficient. Option B reverses these two values, and options C and D use individual numbers rather than the required product and sum. Keeping the sum for the middle coefficient and the product for the constant avoids this common error.

Correct answer: C. ( (x-11)^2 )

(121=11^2) and the middle term is (-22x=-2\times x\times11). Exam tip: identify ((a-b)^2) from the negative middle term.

Correct answer: C. \(56ab\)

Use the identity \((x+y)^2-(x-y)^2=4xy\). Here, \(x=7a\) and \(y=2b\), so the value is \(4\times 7a\times 2b=56ab\). \(49a^2-4b^2\) is the value of \((7a+2b)(7a-2b)\), not the difference of the two squares given here. Exam tip: first identify the pattern \((x+y)^2-(x-y)^2\).

Correct answer: A. \(8p^2+18q^2\)

Using the identity \((a+b)^2+(a-b)^2=2a^2+2b^2\), take \(a=2p\) and \(b=3q\). This gives \(2(2p)^2+2(3q)^2=8p^2+18q^2\). The mixed terms involving \(12pq\) cancel when the two squares are added, so option C is not correct. Exam tip: recognise this identity first to avoid lengthy expansion.

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

The expansion of \((a-b)^2\) is \(a^2-2ab+b^2\), so option A is correct. Option B represents the square of a sum. Exam tip: always check the sign of the middle term.

Correct answer: A. (38809)

The direct answer is option A: 38809. Choose a nearby round number: 197 = 200 - 3. Use (a - b)^2 = a^2 - 2ab + b^2. Thus, 197^2 = (200 - 3)^2 = 200^2 - 2(200)(3) + 3^2 = 40000 - 1200 + 9 = 38809. Option A is correct because the calculation gives 38809. Option B, 38800, is close but leaves out the final +9 or uses an incomplete middle-term calculation. Option C, 39409, is too large and does not follow the subtraction-square identity; using 200 + 3 would give 203^2, not 197^2. Option D, 39809, is also incorrect and does not result from the correct three-term expansion. Another check is ordinary multiplication: 197 × 197 = 197 × (200 - 3) = 39400 - 591 = 38809. The important point is that a number just below a convenient round number can be squared quickly by subtracting twice the product and adding the small square. Memory cue: for (near number - small number)^2, square the near number, subtract the double product, then add the small square.

Correct answer: B. (2xy)

( (x+y)^2=x^2+2xy+y^2 ), so subtracting (x^2+y^2) leaves (2xy). Exam tip: identify the middle term.

Correct answer: B. (-2xy)

( (x-y)^2=x^2-2xy+y^2 ), subtracting (x^2+y^2) gives (-2xy). Exam tip: watch the negative middle term.

Correct answer: B. (2ab+2bc+2ca)

In the square of three terms, the extra mixed terms are (2ab+2bc+2ca). Exam tip: remember all three pairs.

Correct answer: A. \(x^2-5x-36\)

Using the distributive property, \((x+4)(x-9)=x^2-9x+4x-36=x^2-5x-36\). Therefore, option A is correct. In option B, the sign of the middle term is incorrect because \(-9x+4x=-5x\), not \(+5x\). Exam tip: multiply the first, outer, inner and last terms, then combine like terms.

Correct answer: B. (-11)

((-8)+(-3)=-11), so the coefficient of (x) is (-11). Exam tip: the sum of two negative numbers remains negative.

Correct answer: A. \((x+12)(x-12)\)

Since \(144=12^2\), we have \(x^2-144=x^2-12^2\). Using \(a^2-b^2=(a+b)(a-b)\), its factor form is \((x+12)(x-12)\). The expansion of \((x-12)^2\) is \(x^2-24x+144\), so it is not correct. Exam tip: first check whether the constant term is a perfect square.

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

In \((a+b)(a-b)\), the middle terms \(-ab\) and \(+ab\) cancel, leaving \(a^2-b^2\). Options B and C are identities for squaring a binomial. Exam tip: recognise “sum × difference” as a difference of squares.

Correct answer: A. (4ab) and (4ac)

The direct answer is option A: the mixed terms are \(4ab\) and \(4ac\). Use \((p+q+r)^2=p^2+q^2+r^2+2pq+2pr+2qr\), taking \(p=2a\), \(q=b\), and \(r=c\). The three mixed products are \(2(2a)(b)=4ab\), \(2(2a)(c)=4ac\), and \(2(b)(c)=2bc\). Since \(2bc\) is already mentioned, the other two are \(4ab\) and \(4ac\). Option A is therefore correct. Option B gives \(2ab\) and \(2ac\), but it forgets that the first term is \(2a\), not a. Option C lists square terms, not mixed terms, and even \((2a)^2\) equals \(4a^2\). Option D contains an unsuitable three-variable product and a non-mixed linear term. Memory cue: every pair of different terms contributes twice their product.

Correct answer: B. (4)

The expression can be simplified by using the difference of squares. Let the two squares be formed around the same middle term. Direct expansion gives \\(x+5\\)^2=x^2+10x+25 and \\(x-5\\)^2=x^2-10x+25. Subtracting the second expression from the first cancels both \\(x^2\\) and 25, leaving \\(20x\\). The condition therefore becomes \\(20x=80\\).

Dividing both sides by 20 gives \\(x=4\\). Hence option B is correct. The same result follows from \\(a^2-b^2=(a-b)(a+b)\\): here the difference is 10 and the sum is \\(2x\\), so the product is \\(20x\\). Options A, C and D do not satisfy the equation; substituting 4 gives exactly 80. The supplied answer B is therefore verified.

Correct answer: B. (6)

The identity for the square of a sum is \((a+b)^2=a^2+2ab+b^2\). In this question, \(a=x\) and \(b=3\). The middle term is produced by twice the product of the two parts, so it is \(2\times x\times3=6x\). The constant term is the square of 3, which is 9.

Therefore, expanding the left side gives \((x+3)^2=x^2+6x+9\). Comparing this with the given expression \(x^2+kx+9\), the coefficients of the corresponding \(x\)-terms must be equal. Hence \(k=6\), so option B is correct. The value 3 is only the added number, not the coefficient of the middle term. The value 9 is the constant term, and 12 would result from an incorrect multiplication rather than the identity.