Algebraic identities

Apply the identity \((x+y)^2-(x-y)^2=4xy\). Here, \(x=4a\) and \(y=b\), so the expression becomes \(4\times4a\times b=16ab\). \(32ab\) results from introducing an extra factor of 2, so it is incorrect. Exam tip: Recognise this difference-of-squares identity before expanding the brackets.

Use the identity \((a-b)^2=a^2-2ab+b^2\). Taking \(a=3x\) and \(b=5\), we get \((3x-5)^2=9x^2-30x+25\). In option A, the middle term would be \(+30x\), so it is not correct. Exam tip: take the square roots of the first and last terms, then verify the middle term using \(\pm2ab\).

Here, \(9x^2=(3x)^2\) and \(16y^2=(4y)^2\). The middle term is \(2\times3x\times4y=24xy\), so the expression is \((3x+4y)^2\). Exam tip: always verify the middle term.

\(121a^2-49b^2=(11a)^2-(7b)^2\). Applying the difference-of-squares identity \(x^2-y^2=(x+y)(x-y)\) gives \((11a+7b)(11a-7b)\). Options B and C are perfect squares, so their expansions contain a middle term \(\pm154ab\), which is absent in the given expression. Exam tip: first rewrite each term as a perfect square, then check for the difference-of-squares identity.

Expanding the expression,


a(3x-4)(3x+7)=9x^2+21x-12x-28=9x^2+9x-28.

Therefore, the coefficient of (x) is 9. Here, 21 is the coefficient from only one cross-term; the other cross-term, −12x, must also be combined. Exam tip: After multiplying two binomials, always combine like terms.

First expand the product: \((x+4)(x-9)=x^2-9x+4x-36=x^2-5x-36\). Subtracting \(x^2\) gives \(x^2-x^2-5x-36=-5x-36\). Therefore, option B is correct. In option A, the sign of the \(x\)-term is incorrect. Exam tip: while multiplying binomials, check the signs of the middle terms carefully.

Since \((U+V)^2=U^2+2UV+V^2\) and \((U-V)^2=U^2-2UV+V^2\), \(U^2+V^2\) remains unchanged. Only the middle term changes sign. Exam tip: compare the outer terms first.

A has end terms \((4p)^2\), \((3q)^2\) and middle term \(-2(4p)(3q)=-24pq\), so it is \((4p-3q)^2\). B has a negative final term. Tip: check the middle-term sign.

The direct answer is option B: 32000. Use the difference-of-squares identity \(a^2-b^2=(a+b)(a-b)\), rather than calculating two large squares. Here \(a=1008\) and \(b=992\). Thus \(1008^2-992^2=(1008+992)(1008-992)=2000\times16=32000\). Option A, 16000, misses a factor of 2. Option B follows both factors correctly. Option C, 20000, is only the sum factor and does not include the difference factor. Option D, 8000, is also not the product of 2000 and 16. A useful check is that the numbers are close, so their square difference equals their sum times the small difference. Memory cue: difference of squares means add the numbers, subtract the numbers, then multiply.

Write \(997=1000-3\). Using \((a-b)^2=a^2-2ab+b^2\), \((1000-3)^2=1000^2-2\times1000\times3+3^2=1000000-6000+9=994009\). Therefore, option A is correct. \(9940090\) has an extra zero. Exam tip: for squares of numbers close to 1000, use \((a-b)^2\).

\(a^2-b^2=(a+b)(a-b)\) is a difference of squares, so it is not generally the square of a binomial. In contrast, A, B and D are \((a+b)^2\), \((a-b)^2\) and \((2a+b)^2\). Exam tip: use the middle term to identify the binomial signs.

In \((x+y+z)^2\), the square terms appear along with twice each pairwise product. Hence A is correct. In B, the mixed-term coefficients are 1, not 2. Exam tip: check every mixed-term coefficient.

Use the identity \\( (u-v)^2-(u+v)^2=-4uv\\). Here, u=5m and v=2n. The first square has the minus form and the second square has the plus form, so their difference is negative. Substituting gives \\(-4(5m)(2n)=-40mn\\). Therefore option B is correct. The order of subtraction matters; changing it would change the sign of the result.

This can also be checked by expansion. The first square is \\(25m^2-20mn+4n^2\\), and the second is \\(25m^2+20mn+4n^2\\). Subtracting the second from the first cancels the two square terms and gives \\(-20mn-20mn=-40mn\\). Thus neither 40mn nor 20mn has the correct sign or magnitude. The expressions 25m² and 4n² are only parts of the expansion, not the final simplified difference.

The correct identity is \(a^2-b^2=(a-b)(a+b)\). On multiplying the factors, the middle terms cancel, leaving \(a^2-b^2\). Option B is incorrect because \((a-b)^2\) contains the middle term \(-2ab\). Exam tip: look for paired factors with opposite signs.

Here \(9a^2=(3a)^2\) and \(16b^2=(4b)^2\), while the middle term is \(-2\times3a\times4b=-24ab\). So it is \((3a-4b)^2\). In A, the middle term should be \(8x\), not \(10x\). Exam tip: verify the middle term using \(\pm2ab\).

Multiply the first two factors: \((x+6)(x-4)=x^2-4x+6x-24=x^2+2x-24\). Therefore, option A is correct. In option B, the middle term is incorrect, while option D has the wrong sign for the constant term. Exam tip: when multiplying two binomials, check the sum of the inner and outer terms for the middle term.

Expanding \((a+b)^2\) gives \(a^2+2ab+b^2\), so its \(ab\)-term is \(2ab\). The mixed terms in \((b+c)^2\) and \((c+a)^2\) are \(2bc\) and \(2ca\), respectively, not \(ab\). Therefore, the total \(ab\)-term is \(2ab\). Exam tip: Expand each square separately and collect only the requested term.

For the first part, \((x+y)^2-(x-y)^2=4xy\). Similarly, \((x+2y)^2-(x-2y)^2=8xy\). Therefore, the total is \(4xy+8xy=12xy\). It is not \(16xy\), because the two differences add up to 12. Exam tip: apply the identity \((a+b)^2-(a-b)^2=4ab\) directly.

Use the identity \((a-b)(a+b)=a^2-b^2\). Here, \(a=2x\) and \(b=5\), so \((2x-5)(2x+5)=4x^2-25\). Adding \(25\) gives \(4x^2-25+25=4x^2\). The option \(4x^2+25\) incorrectly fails to cancel the \(-25\) with the outside \(+25\). Exam tip: When conjugate factors appear, apply the difference-of-squares identity first.

The expression matches the identity \(a^2+b^2-2ab=(a-b)^2\), where \(a=x+3\) and \(b=x+4\). Therefore, its value is \(\big((x+3)-(x+4)\big)^2=(-1)^2=1\). It would be 0 only if the two terms were equal; here they differ by 1. Exam tip: First identify \(a^2+b^2-2ab\) and rewrite it as \((a-b)^2\).

Use the identity \((x+y)^2-(x-y)^2=4xy\). Here, \(x=3a\) and \(y=2b\), so the expression equals \(4\times 3a\times 2b=24ab\). Hence, option B is correct. \(9a^2-4b^2\) is the product \((3a+2b)(3a-2b)\), whereas the question gives the difference of the squares of these two binomials. Exam tip: for an expression of the form \((x+y)^2-(x-y)^2\), apply \(4xy\) directly.

\(x^2+4x+4=(x+2)^2\). The square of every real number is never negative, so this polynomial is always non-negative. In contrast, \(x^2-4\) is negative at \(x=0\). Exam tip: check whether the middle term is twice the product of the terms in the binomial.

In conjugate binomials, one term is the same and the other has an opposite sign. Thus, \((5x+2y)(5x-2y)=25x^2-4y^2\). In option B both binomials are identical, so it uses a square identity instead. Exam tip: check for matching terms and opposite signs.

We use \( (x+y+z)^2=x^2+y^2+z^2+2(xy+yz+zx) \). Substitute \(x=1\), \(y=2\), \(z=3\): \(x^2+y^2+z^2=1^2+2^2+3^2=1+4+9=14\). Also \(xy+yz+zx=1\cdot2+2\cdot3+3\cdot1=2+6+3=11\). Therefore \((x+y+z)^2=14+2\cdot11=14+22=36\).

\(p^2+2pq+q^2=(p+q)^2\). The square of every real number is zero or positive, so this expression cannot be negative. Exam tip: check whether the middle term is \(2pq\) to spot a perfect square.