Algebraic identities

Use the difference-of-squares identity \\( (u+v)^2-(u-v)^2=4uv\\). In this expression, \\(u=x\\) and \\(v=5\\). Hence the value is \\(4(x)(5)=20x\\). Direct expansion gives \\(x^2+10x+25-(x^2-10x+25)\\); the equal square and constant terms cancel, leaving \\(20x\\). Both methods confirm the same result.

Therefore option C is correct. Option A, \\(10x\\), keeps only one of the two linear contributions, while option B incorrectly treats the expression as a product involving 25x. Option D ignores the variable terms and is not obtained by the identity. Recognising the plus-minus square pattern makes the calculation quick and reduces sign errors.

The direct answer is option D, \(2a^2+2b^2\). Start with the two standard identities: \((a+b)^2=a^2+2ab+b^2\) and \((a-b)^2=a^2-2ab+b^2\). Add them carefully: \(a^2+2ab+b^2+a^2-2ab+b^2\). The middle terms \(+2ab\) and \(-2ab\) cancel, leaving \(2a^2+2b^2\). Option A, \(4ab\), is not the sum; it is related to the difference of the two squares. Option B, \(2ab\), contains only the middle-term part and misses both square terms. Option C, \(a^2+b^2\), forgets that each square term occurs twice. Option D includes both doubled square terms and is correct. This identity is useful because expansion becomes quick and orderly. Memory cue: adding the plus-square and minus-square makes the middle terms disappear.

In the identity \((a+b)^2=a^2+2ab+b^2\), the last term is \(b^2\). Here, \(a=2x\) and \(b=3y\), so the last term is \((3y)^2=9y^2\). \(12xy\) is the middle term because \(2(2x)(3y)=12xy\). Exam tip: in a square expansion, the first and last terms are the squares of the two individual terms.

\(x^2-9=x^2-3^2\), so it matches \(a^2-b^2\) and factorises as \((x+3)(x-3)\). \(x^2+9\) is a sum of squares, not a difference. Exam tip: identify perfect squares first.

Using the distributive property, \((m+4)(m+6)=m^2+6m+4m+24=m^2+10m+24\). Hence, option A is correct. In option B, the middle-term coefficient and constant term are interchanged, while option C incorrectly adds \(6m+4m\). Exam tip: in \((m+a)(m+b)\), the middle term is \((a+b)m\) and the constant term is \(ab\).

\((x-2)(x-5)=x^2-7x+10\). Hence, the term without \(x\), i.e. the constant term, is \(10\). Here, \(-7\) is the coefficient of \(x\), not the constant term. Exam tip: in the product of two binomials, multiply the constant terms to find the constant term.

Using the distributive property, \((x-2)(x-5)=x^2-5x-2x+10=x^2-7x+10\). Therefore, the coefficient of \(x\) is \(-7\). The number \(10\) is the constant term, not the coefficient of \(x\). Exam tip: in \((x-a)(x-b)\), the coefficient of \(x\) is \(-(a+b)\).

This expression uses the identity \((a+b)(a-b)=a^2-b^2\). Taking \(a=z\) and \(b=9\), \((z+9)(z-9)=z^2-9^2=z^2-81\). \(z^2+81\) is incorrect because this is a difference of squares, not a sum. Exam tip: for binomials with opposite signs, the middle terms cancel.

The direct answer is A: the other two product terms are 2ac and 2bc. Expand by multiplying every term in the first bracket by every term in the second: \\(a+b+c)^2=(a+b+c)(a+b+c)\\. The square terms are a², b² and c². Each unlike pair occurs twice: ab+ab=2ab, ac+ac=2ac, and bc+bc=2bc. Therefore the full identity is \\(a^2+b^2+c^2+2ab+2ac+2bc\\. Option A is correct. B gives square terms, not the two other pairwise products. C misses the coefficient 2 and lists only two products. D contains an incorrect 3abc term; a squared trinomial has no such cubic term, although c² is present. Exam cue: with three terms, remember three pairs—ab, ac, bc—and double each.

A square term is produced when a quantity is multiplied by itself. In the expression \\( (x+y+z)^2 \\), the individual quantities are squared, giving \\(x^2\\), \\(y^2\\), and \\(z^2\\). These are called the square terms. When the expression is expanded, it also contains the mixed products \\(2xy\\), \\(2yz\\), and \\(2zx\\), but those are not square terms because they involve two different variables.

The identity is \\( (x+y+z)^2=x^2+y^2+z^2+2xy+2yz+2zx\\). Thus option B lists exactly the three terms in which each variable is squared. Option A lists unsquared products, option C lists doubled cross-products, and option D is the original sum rather than a list of terms. The distinction is based on the exponent and the variables present.

The square of three terms has three square terms and three double product terms. Exam tip: write all terms with (2).

The square of three terms contains the square of each term and twice the product of every distinct pair. For the terms in \\(x+2+y\\), the pair consisting of \\(x\\) and \\(y\\) contributes \\(2xy\\). Other pairwise products also occur, such as \\(4x\\) and \\(4y\\), along with \\(x^2\\), \\(4\\), and \\(y^2\\).

Thus, option B is correct. The term \\(2xy\\) does not appear because x and y are square terms; it appears because their product is counted twice when the whole sum is multiplied by itself. The number 2 does not disappear, and the expansion definitely contains product terms. Pairing the three terms prevents omissions.

Using the identity \((a+b)^2=a^2+2ab+b^2\), the given expression is \((a+b)^2\); hence option C is correct. In \((a-b)^2\), the middle term is \(-2ab\), so it cannot match. Exam tip: use the sign of the middle term to distinguish between the two square identities.

On expanding \((a+b)^2\), the terms obtained are \(a^2\), \(2ab\), and \(b^2\), so option A is correct. Option B is the identity for the square of a difference. Exam tip: the middle term is positive in the square of a sum.

Expanding \((a+b)^2\) gives \(a^2+2ab+b^2\), so option A is correct. In \((a-b)^2\), the middle term is \(-2ab\). Exam tip: check the sign of the middle term first.

\(x^2-25=x^2-5^2\), so it has the form \(a^2-b^2\). In contrast, \(x^2+25\) is a sum of squares. Exam tip: check whether each term is a perfect square.

\(x^2-9=x^2-3^2\), so it is a difference of two perfect squares and factors as \((x-3)(x+3)\). \(x^2+9\) is a sum of squares. Exam tip: first check whether both terms are perfect squares.

\(x^2-y^2\) is the difference of two perfect squares and factors as \((x-y)(x+y)\). Options B and D are perfect-square trinomials. In exams, check for two square terms separated by a minus sign.

The direct answer is A. The first expression is a squared binomial: \\(x-1\\)^2=x^2-2x+1\\). The second is a difference of squares: \\(x^2-1=(x-1)(x+1)\\). Step by step, squaring \\(x-1\\) means multiplying \\(x-1\\) by itself, which gives \\(x^2-x-x+1=x^2-2x+1\\). But \\(x^2-1\\) factors using \\(a^2-b^2=(a-b)(a+b)\\), with \\(a=x,b=1\\). Option A is correct. B is wrong because the expressions are not always equal; for example, at \\(x=2\\), they are \\(1\\) and \\(3\\). C wrongly assigns the expansion of the first expression to the second. D has the wrong sign: \\(x^2+2x+1\\) equals \\( (x+1)^2\\). Memory cue: minus inside a square gives a negative middle term.

This expression has the form \((p+q)(p-q)=p^2-q^2\). Here, \(p=2a\) and \(q=3\), so \((2a+3)(2a-3)=(2a)^2-3^2=4a^2-9\). Option A incorrectly adds the two squares. Exam tip: for conjugate binomials, the middle terms cancel, leaving the difference of the squares.

In \((a+b)(a-b)\), the middle terms \(+ab\) and \(-ab\) cancel, leaving \(a^2-b^2\). Option C also contains the term \(-2ab\). Exam tip: conjugate binomials give the difference of two squares.

The identity is \((a+b)^2=a^2+2ab+b^2\). Its middle term is twice the product of the two terms. Option B represents \((a-b)^2\). Exam tip: always check the sign of the middle term.

\((a+b)^2=(a+b)(a+b)\). Multiplying gives \(a^2+ab+ab+b^2=a^2+2ab+b^2\). Option B is the expansion of \((a-b)^2\). Exam tip: check the sign of the middle term in square identities.

In both, (x^2) and (64) remain same; only the sign of the middle term changes. Exam tip: compare signs.

The direct answer is option C: identities are mainly used to expand and simplify algebraic expressions quickly. An identity is an equality that remains true for all permitted values of its variables, such as \((a+b)^2=a^2+2ab+b^2\). Instead of multiplying every term separately, we recognise the pattern and apply the known identity. Option A, only drawing figures, belongs mainly to geometry and is not the central purpose of algebraic identities. Option B, only reading fractions, has no connection with these formulas. Option C correctly describes both important jobs: expansion changes a compact product or square into terms, while simplification collects terms into an easier form. Option D, only measuring angles, is a geometry activity and cannot describe algebraic identities. These identities save time and reduce calculation errors, but the pattern must be matched correctly. Memory cue: see a familiar algebraic shape, name its identity, then substitute carefully.