Algebraic identities

Correct answer: C. \(4\)

Using \((a+b)^2=a^2+2ab+b^2\), we get \((5x+2)^2=25x^2+20x+4\). A term with no \(x\) is called the constant term, so the constant term is \(4\). \(20x\) is the middle term and is not constant. Exam tip: after expanding, identify the term without the variable.

Correct answer: B. (-8p)

The direct answer is option B: -8p. Start with the identity (x - y)^2 = x^2 - 2xy + y^2. Here x = 4p and y = 1. Therefore, (4p - 1)^2 = (4p)^2 - 2(4p)(1) + 1^2 = 16p^2 - 8p + 1. The term containing p to the first power is -8p. Option A, 8p, has the wrong sign because the middle term of a subtraction square is negative. Option B, -8p, is correct because -2 multiplied by 4p and 1 gives -8p. Option C, 16p^2, is also part of the expansion, but it contains p squared, not p alone, so it is not the requested term. Option D, 1, is the constant term and contains no p. The important lesson is that the middle term is formed by multiplying twice the first part and the second part, with the sign between the parts retained. Memory cue: in (a - b)^2, remember square, minus double product, square.

Correct answer: A. ( (1000+2)^2 )

The number 1002 is very close to the simple number 1000, so the most useful rewrite is \\(1002=1000+2\\). Hence \\(1002^2=(1000+2)^2\\). Using \\((a+b)^2=a^2+2ab+b^2\\), this can be calculated as \\(1000^2+2(1000)(2)+2^2=1,000,000+4,000+4=1,004,004\\). The question asks for the useful form, not merely the final numerical value.

Therefore option A is correct. The form \\(1000-2\\) represents 998, not 1002, so option B is unsuitable. Similarly, \\(100+2\\) is 102 and \\(10+2\\) is 12; their squares are different numbers. The key strategy is to express a number near a convenient base and then apply the square identity. This reduces long multiplication and lowers the chance of calculation errors.

Correct answer: B. ( (1000-1)^2 )

The correct answer is option B: \((1000-1)^2\). The number 999 is one less than the convenient base 1000, so write \(999=1000-1\). Therefore \(999^2=(1000-1)^2\). Using \((a-b)^2=a^2-2ab+b^2\), we get \(999^2=1000000-2000+1=998001\). Option A, \((900+99)^2\), is numerically equal to 999 squared, but it is not the intended convenient identity form and is less efficient. Option B is the best and correct form because it uses a nearby round number. Option C represents \(1001^2\), not \(999^2\). Option D represents \(108^2
eq999^2\). Memory cue: numbers just below a power of 10 are written as base minus 1.

Correct answer: C. \(uv\)

Using the identity \((u+v)(u-v)=u^2-v^2\), multiplication gives \(uv\) and \(-uv\), which cancel each other. Therefore, the term \(uv\) does not appear in the final expression. The term \(-v^2\) does appear, so it is not the correct choice. Exam tip: recognise \((a+b)(a-b)\) directly as \(a^2-b^2\).

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

In \((a+b)(a-b)\), the middle terms \(+ab\) and \(-ab\) cancel, leaving \(a^2-b^2\). Option B is the identity for the square of a sum. Exam tip: look for conjugate binomials with opposite signs.

Correct answer: C. \(121\)

Using the identity \((a-b)^2=a^2-2ab+b^2\), we get \((n-11)^2=n^2-22n+121\). Therefore, the last term is \(121\). \(-22n\) is the middle term; the last term is \(11^2\), which is positive. Exam tip: in \((a-b)^2\), the last term is always \(b^2\).

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

\(x^2+10x+25=x^2+2\cdot x\cdot5+5^2=(x+5)^2\), so it is a perfect-square trinomial. In option B, the constant would need to be \(25\). Exam tip: match the middle term with \(2ab\).

Correct answer: A. (9x^2-12xy+4y^2)

The direct answer is A: \\(3x-2y\\)^2=9x^2-12xy+4y^2. Start with the identity \\(a-b\\)^2=a^2-2ab+b^2. Put \\(a=3x\\) and \\(b=2y\\). Then \\(a^2=(3x)^2=9x^2\\); the middle term is \\(-2(3x)(2y)=-12xy\\); and \\(b^2=(2y)^2=4y^2\\). Combining them gives \\(9x^2-12xy+4y^2\\). Option A has all three correct terms and the correct negative middle sign. B is wrong because the middle term has a plus sign, which belongs to \\(a+b\\)^2\\). C is wrong because the first and last squares have incorrect coefficients. D is wrong because it omits the middle term and represents a different difference-of-squares expression. Memory cue: for \\(a-b\\)^2\\), square the first, subtract twice the product, then square the second.

Correct answer: B. \(25m^2-16n^2\)

This uses the identity \((a+b)(a-b)=a^2-b^2\). Here, \(a=5m\) and \(b=4n\), so \((5m+4n)(5m-4n)=(5m)^2-(4n)^2=25m^2-16n^2\). Option D resembles the expansion of \((5m-4n)^2\), but the given factors have opposite signs. Exam tip: in conjugate factors, the middle terms cancel out.

Correct answer: A. ( (x+9)^2 )

A perfect square trinomial has the form \\(x^2+2kx+k^2=(x+k)^2\\). In the given expression, the last term is \\(81=9^2\\). Twice the product of \\(x\\) and 9 is \\(2(x)(9)=18x\\), which matches the middle term. Therefore \\(x^2+18x+81=(x+9)^2\\). Substitution or expansion can be used to verify this result.

Option A is correct. Option B would expand to \\(x^2-18x+81\\), so its middle-term sign is wrong. Options C and D use numbers that do not produce both the constant term and the coefficient of \\(x\\). A reliable method is to take the square root of the constant term and then check whether twice that number gives the middle coefficient.

Correct answer: B. ( (k-10)^2 )

The direct answer is B:
\(k^2-20k+100=(k-10)^2\). Start with the identity \((a-b)^2=a^2-2ab+b^2\). Here, take \(a=k\) and \(b=10\). Then \((k-10)^2=k^2-2(k)(10)+10^2=k^2-20k+100\), exactly the given expression. Therefore its square root expression is \((k-10)\). Option A, \((k+20)^2\), would give \(k^2+40k+400\), so both the middle term and constant are wrong. Option B gives the exact expansion, so it is correct. Option C, \((k+10)^2\), gives \(k^2+20k+100\); the sign of the middle term is wrong. Option D, \((k-100)^2\), gives \(k^2-200k+10000\), which is also different. The important clue is that the last term is \(100=10^2\), and the middle term is negative \(-20k=-2(k)(10)\). Memory trick: a negative middle term usually indicates the form \((a-b)^2\), while a positive middle term indicates \((a+b)^2\). Always expand the proposed square to check it.

Correct answer: C. (640)

Use the difference-of-squares identity \\(a^2-b^2=(a+b)(a-b)\\). The expression is \\(37^2-27^2\\), so set \\(a=37\\) and \\(b=27\\). Their sum is \\(37+27=64\\), and their difference is \\(37-27=10\\). Thus \\(37^2-27^2=64\\times10=640\\). This avoids separately finding 1369 and 729 and then subtracting them.

Therefore, option C, 640, is correct. A quick direct check is \\(37^2=1369\\) and \\(27^2=729\\); subtracting gives \\(1369-729=640\\). The other values are not equal to the difference. Recognising two squared numbers separated by a minus sign is the essential step in this problem.

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

The numbers 45 and 55 are equally distant from 50: they can be written as \\(50-5\\) and \\(50+5\\). Their product therefore has the form \\((a-b)(a+b)\\), which uses the identity \\((a+b)(a-b)=a^2-b^2\\). Taking \\(a=50\\) and \\(b=5\\), we get \\(45\\times55=(50-5)(50+5)=50^2-5^2\\). The identity named in option C is exactly this rule.

Thus option C is correct. If the value is required, it is \\(2500-25=2475\\). Options A and B are square identities but do not directly represent the product of two conjugate binomials. Option D is an identity for three terms and is unnecessary here. The useful pattern is two numbers equally placed around a common centre.

Correct answer: B. From \((2b)^2\)

In the expansion of a three-term square, the square of each individual term appears. Here, \((2b)^2=4b^2\), so option B is correct. \(2ac\) is a product term involving \(a\) and \(c\), not the source of \(4b^2\). Exam tip: when squaring a term, square its numerical coefficient as well.

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

The correct identity is \(a^2-b^2=(a+b)(a-b)\), because multiplying the factors cancels the middle terms. Option C misses the \(-2ab\) term. Exam tip: recognise a difference of squares and factor it into sum and difference.

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

Use the identity \((a-b)(a+b)=a^2-b^2\). Here, \((x-3)(x+3)=x^2-3^2=x^2-9\). Therefore, \((x^2-9)+9=x^2\), so option A is correct. \(x^2-9\) is only the product and does not include the outside \(+9\). Exam tip: when you see conjugate binomials \((a-b)(a+b)\), apply the difference-of-squares identity directly.

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

The difference-of-squares identity is \(a^2-b^2=(a-b)(a+b)\). Expanding gives \(a^2+ab-ab-b^2=a^2-b^2\). Option B introduces an extra \(-2ab\) term. Exam tip: use this identity when two squared terms are separated by a minus sign.

Correct answer: A. ( (x+7)^2 )

When the same algebraic factor is multiplied by itself, the product is its square: \\(A\cdot A=A^2\\). Here the repeated factor is \\(x+7\\), so \\((x+7)(x+7)=(x+7)^2\\). Expanding it would give \\(x^2+14x+49\\), which is also consistent with the identity \\( (x+y)^2=x^2+2xy+y^2\\) for \\(y=7\\).

Therefore option A is correct. Option B changes the plus sign to a minus sign and represents a different expression. Options C and D correspond to the difference of squares, \\((x+7)(x-7)=x^2-49\\), not to two identical factors. The essential recognition is that repeated multiplication of one expression is written as a power of 2.

Correct answer: B. ( (a-b)^2 ) has (-2ab) term but (a^2-b^2) does not

( (a-b)^2=a^2-2ab+b^2 ) while (a^2-b^2=(a+b)(a-b)). Exam tip: keep square of bracket and difference of squares separate.

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

\((x+y)^2=(x+y)(x+y)\). On multiplying, the middle terms give \(xy+xy=2xy\), so the expression is \(x^2+2xy+y^2\). Exam tip: check the sign of the middle term carefully.

Correct answer: B. \((6+x)(6-x)\)

This is a difference of squares: \(36-x^2=6^2-x^2\). Using \(a^2-b^2=(a+b)(a-b)\), we get \((6+x)(6-x)\). Option A expands to \(x^2-36\), which is the negative of the given expression. Exam tip: first identify the square roots of both terms, then write one sum factor and one difference factor.

Correct answer: A. (10)

The direct answer is option A: the coefficient of \(x\) is 10. Multiply each term in the first bracket by each term in the second bracket: \((x+4)(x+6)=x\cdot x+x\\cdot6+4\cdot x+4\cdot6\). This gives \(x^2+6x+4x+24=x^2+10x+24\). The coefficient is the number multiplying \(x\), so it is 10. Equivalently, the linear coefficient comes from adding the constants 4 and 6. Option A is correct. Option B, 24, is the constant term because it comes from \(4\cdot6\), not the coefficient of \(x\). Option C, 4, is only one contribution to the linear term. Option D, 6, is the other contribution, but both contributions must be added. Do not confuse a term with its coefficient: in \(10x\), the coefficient is 10, while 10x is the complete term. Memory cue: for \((x+p)(x+q)\), the middle coefficient is \(p+q\), and the constant term is \(pq\).

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

The difference-of-squares identity is \((a+b)(a-b)=a^2-b^2\). On expansion, the middle terms cancel. Exam tip: a product of a sum and a difference gives a difference of squares.

Correct answer: B. (12m)

The expression has the form
\((a+b)^2=a^2+2ab+b^2\). The middle term is the product term in the centre, formed by twice the product of the two terms being added. Here, those terms are \(2m\) and \(3\). Therefore, the middle term is \(2(2m)(3)=12m\). Thus, option B is correct. Option C, \(4m^2\), is the first square, and option D, \(9\), is the last square; neither is the middle term.

To solve similar questions, first identify \(a\) and \(b\), then use \(2ab\). Substituting gives \(2\times2m\times3=12m\). The coefficient is multiplied by both the variable term and the constant, so stopping at \(6m\) misses the required factor of 2. The supplied answer and explanation are mathematically correct and clearly identify the middle term.