Question
Expand \((3x-2y)^2\)
Working
\((3x)^2-2(3x)(2y)+(2y)^2=9x^2-12xy+4y^2\) Trick: square the first and last terms; the middle term is twice their product
सर्वसमिकाएंAlgebraic Identities
Core identities, short proofs and worked exam questions.
1. Squares of a sum and a difference
\((a+b)^2=a^2+2ab+b^2\)
\((a-b)^2=a^2-2ab+b^2\)
Derivation:
\((a+b)^2=(a+b)(a+b)=a^2+ab+ab+b^2=a^2+2ab+b^2\). Similarly, \((a-b)^2=(a-b)(a-b)=a^2-ab-ab+b^2=a^2-2ab+b^2\). The last term is positive in both identities.
Question
Find the missing term: \((2p+5q)^2=4p^2+\square+25q^2\)
Working
\(\square=2(2p)(5q)=20pq\)
Question
Find \(k\) if \((x+k)^2=x^2+10x+\square\)
Working
Compare the middle term: \(2kx=10x\), so \(k=5\); the missing constant is \(25\)
2. Conjugates and difference of squares
\((a+b)(a-b)=a^2-b^2\).
The reverse form is \(a^2-b^2=(a-b)(a+b)\). Repeated application gives \(a^4-b^4=(a-b)(a+b)(a^2+b^2)\).
Derivation:
\((a+b)(a-b)=a^2-ab+ab-b^2=a^2-b^2\); the cross terms cancel.
Question
Expand \((5x+3y)(5x-3y)\)
Working
\((5x)^2-(3y)^2=25x^2-9y^2\)
Question
Factorise completely \(x^4-81\)
Working
\((x^2-9)(x^2+9)=(x-3)(x+3)(x^2+9)\) Trick: keep factoring any difference of squares; a sum of positive squares does not factor over the real numbers
Question
Simplify \((m^2-n^2)/(m-n)\)
Working
\(((m-n)(m+n))/(m-n)=m+n\), provided \(m\ne n\) Trick: cancel factors, never individual terms
3. Product of two binomials
\((x+a)(x+b)=x^2+(a+b)x+ab\).
More generally, \((px+q)(rx+s)=prx^2+(ps+qr)x+qs\). For three factors, \((x+a)(x+b)(x+c)=x^3+(a+b+c)x^2+(ab+bc+ca)x+abc\).
Derivation:
\((x+a)(x+b)=x^2+bx+ax+ab=x^2+(a+b)x+ab\). For three factors, \((x+a)(x+b)(x+c)=[x^2+(a+b)x+ab](x+c)=x^3+(a+b+c)x^2+(ab+ac+bc)x+abc\).
Question
Expand \((x+7)(x-3)\)
Working
\(x^2+(7-3)x+(7)(-3)=x^2+4x-21\) Trick: add the constants for the x coefficient, multiply them for the constant
Question
Expand \((2x+3)(3x-4)\)
Working
\(6x^2-8x+9x-12=6x^2+x-12\)
Question
Factorise \(x^2-x-20\)
Working
\((-5)(4)=-20\), \(-5+4=-1\); hence \((x-5)(x+4)\)
Question
Expand \((x+1)(x+2)(x+3)\)
Working
First multiply two factors: \((x+1)(x+2)=x^2+3x+2\); then \((x^2+3x+2)(x+3)=x^3+6x^2+11x+6\).
4. Perfect-square trinomials and three-term squares
\(a^2+2ab+b^2=(a+b)^2\), \(a^2-2ab+b^2=(a-b)^2\).
For three terms: \((a+b+c)^2=a^2+b^2+c^2+2ab+2bc+2ca\).
Derivation of the three-term result: \((a+b+c)(a+b+c)\); distribute each term to obtain \(a^2+b^2+c^2+ab+ab+bc+bc+ca+ca\), then combine like terms.
Question
Factorise \(9x^2-30xy+25y^2\)
Working
\((3x)^2-2(3x)(5y)+(5y)^2=(3x-5y)^2\)
Question
Expand \((x+2y-3z)^2\)
Working
\(x^2+4y^2+9z^2+4xy-6xz-12yz\) Trick: write all three squares and twice each pairwise product, keeping each sign
Question
Find \(c\) if \(4x^2+cxy+9y^2=(2x+3y)^2\)
Working
Expand the right side: \(4x^2+12xy+9y^2\); therefore \(c=12\)
5. Cubes of a sum and a difference
\((a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)\).
\((a-b)^3=a^3-3a^2b+3ab^2-b^3=a^3-b^3-3ab(a-b)\).
Derivation:
\((a+b)^3=(a+b)(a^2+2ab+b^2)\); distributing gives \(a^3+3a^2b+3ab^2+b^3\). Replacing \(b\) by \(-b\) gives the difference formula.
Question
Expand \((2x+y)^3\)
Working
\(8x^3+12x^2y+6xy^2+y^3\) Trick: the coefficients are 1, 3, 3, 1; keep the powers of the first term descending
Question
Expand \((3p-2q)^3\)
Working
\(27p^3-54p^2q+36pq^2-8q^3\) Trick: signs alternate for a difference cube; the middle two terms do not both become negative
Question
Evaluate \(101^3\)
Working
\((100+1)^3=1000000+30000+300+1=1030301\)
6. Factorising sums and differences of cubes
\(a^3+b^3=(a+b)(a^2-ab+b^2)\).
\(a^3-b^3=(a-b)(a^2+ab+b^2)\).
Derivation:
\((a+b)(a^2-ab+b^2)=a^3+b^3\); the mixed terms cancel. Similarly, \((a-b)(a^2+ab+b^2)=a^3-b^3\).
Question
Factorise \(x^3+8\)
Working
\(x^3+2^3=(x+2)(x^2-2x+4)\)
Question
Factorise \(27a^3-64b^3\)
Working
\((3a)^3-(4b)^3=(3a-4b)(9a^2+12ab+16b^2)\) Trick: the quadratic factor has the same-sign middle term as the first factor’s opposite sign
Question
Solve \(x^3-125=0\)
Working
Factorise: \((x-5)(x^2+5x+25)=0\). Its discriminant is \(5^2-4(1)(25)=-75<0\)
7. Three-variable identities and related products
\(a^3+b^3+c^3-3abc=(a+b+c)(a^2+b^2+c^2-ab-bc-ca)\).
Therefore, if \(a+b+c=0\), then \(a^3+b^3+c^3=3abc\). Also, \((a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc\). The three-term cube is \((a+b+c)^3=a^3+b^3+c^3+3(a+b+c)(ab+bc+ca)-3abc\); expanding the product gives the six pairwise terms and \(6abc\).
Derivation of the sum-of-cubes identity. Put \(Q=a^2+b^2+c^2-ab-bc-ca\). Distribute each term:
\(aQ=a^3+ab^2+ac^2-a^2b-abc-a^2c\)
\(bQ=a^2b+b^3+bc^2-ab^2-b^2c-abc\)
\(cQ=a^2c+b^2c+c^3-ac^2-bc^2-abc\)
Adding these lines cancels every mixed pair and leaves \((a+b+c)Q=a^3+b^3+c^3-3abc\). If \(a+b+c=0\), the left side is zero; hence \(a^3+b^3+c^3=3abc\).
For the pair-sum identity, expand: \((a+b)(b+c)(c+a)=a^2b+a^2c+ab^2+ac^2+b^2c+bc^2+2abc\). Also \((a+b+c)(ab+bc+ca)\) is the same six mixed terms plus \(3abc\); subtracting \(abc\) gives the identity.
For the three-term cube, multiply its square by the sum: \((a+b+c)^3=(a+b+c)(a^2+b^2+c^2+2ab+2bc+2ca)\). Distributing and collecting gives the stated compact form.
Question
Evaluate \(x^3+y^3+z^3-3xyz\) when \(x+y+z=0\)
Working
The factor \(x+y+z\) is zero, so the value is \(0\)
Question
Find \(a^3+b^3+c^3\) if \(a+b+c=0\) and \(abc=12\)
Working
Use \(a^3+b^3+c^3=3abc=36\)
Question
Evaluate \((a+b)(b+c)(c+a)\) for \(a=1,b=2,c=3\)
Working
\((3)(5)(4)=60\); checking the right side gives \((6)(11)-6=60\)
Question
Expand \((x+y+1)^3\)
Working
Use the three-term cube: \(x^3+y^3+1+3(x^2y+x^2+y^2x+y^2+x+y)+6xy\).
8. Derived forms, substitutions and quick calculation
\((a+b)^2+(a-b)^2=2(a^2+b^2)\); \((a+b)^2-(a-b)^2=4ab\); \(a^2+b^2=(a+b)^2-2ab\); \((a-b)^2=(a+b)^2-4ab\).
The identities follow by adding or subtracting the two square expansions; they let you work with a sum and product without finding the numbers separately.
Question
Find \(a^2+b^2\) and \((a-b)^2\) when \(a+b=13\) and \(ab=30\)
Working
\(a^2+b^2=13^2-2(30)=109\); \((a-b)^2=13^2-4(30)=49\)
Question
Calculate \(998\times1002\)
Working
\((1000-2)(1000+2)=1000^2-2^2=999996\) Trick: express the numbers as conjugates around the same base
Question
Simplify \((x+4)^2-(x-4)^2\)
Working
\([(x+4)-(x-4)][(x+4)+(x-4)]=(8)(2x)=16x\)
9. Exam question forms: prove, complete, compare and detect errors
Questions may ask you to expand, factorise fully, match an identity, fill a missing term, find an unknown coefficient, compare two expressions, simplify a quotient, evaluate from given symmetric information, solve an equation, prove an identity, identify an incorrect step, or apply an identity to a number or simple area. Always show the expression being asked about and preserve its domain restrictions.
Question
Complete \(x^2+14x+\square=(x+\square)^2\)
Working
Half of 14 is 7, so \(x^2+14x+49=(x+7)^2\)
Question
Find all real \(x\) satisfying \(x^2-16=0\)
Working
\((x-4)(x+4)=0\); hence \(x=4\) or \(x=-4\)
Question
Find the error in \((x-3)^2=x^2-9\)
Working
The cross term was omitted. Correct expansion: \((x-3)^2=x^2-6x+9\)
Question
Prove \((a+b)^2-(a-b)^2=4ab\)
Working
Expand both squares: \((a^2+2ab+b^2)-(a^2-2ab+b^2)=4ab\)
Exam check:
substitute simple values into a claimed identity as a quick error check, then verify by expansion; a numerical check alone is not a proof. For rational expressions, state excluded values before cancelling.
Question
Find the area of a square with side \((x+3)\) cm
Working
\(A=(x+3)^2=x^2+6x+9\) cm².
10. Competitive MCQs: multi-step identity problems
Choose one option before reading the reasoning. These questions combine identities, factorisation, domain restrictions and algebraic shortcuts; several distractors reflect common sign or cancellation errors.
Question · MCQ
Evaluate \((1003^2-997^2)/6\).
Correct option and reasoning
Correct option: B. \(1003^2-997^2=(1003-997)(1003+997)=6\cdot2000\); dividing by 6 gives \(2000\).
Question · MCQ
Find \(103^3+97^3\).
Correct option and reasoning
Correct option: A. Write \(103=100+3\) and \(97=100-3\). The sum is \(2(100)^3+6(100)(3)^2=2005400\); the odd terms cancel.
Question · MCQ
If \(x+1/x=3\), find \(x^3+1/x^3\).
Correct option and reasoning
Correct option: A. \(u^3+v^3=(u+v)^3-3uv(u+v)\); take \(u=x, v=1/x\), so \(uv=1\) and the value is \(3^3-3(1)(3)=18\).
Question · MCQ
Given \(a+b+c=6\), \(ab+bc+ca=11\) and \(abc=6\), find \(a^3+b^3+c^3\).
Correct option and reasoning
Correct option: A. From the three-term cube identity, \(a^3+b^3+c^3=(a+b+c)^3-3(a+b+c)(ab+bc+ca)+3abc\). Thus \(216-198+18=36\).
Question · MCQ
If \(a+b=7\) and \(ab=10\), find \(a^4+b^4\).
Correct option and reasoning
Correct option: A. First \(a^2+b^2=7^2-2(10)=29\). Then \(a^4+b^4=(a^2+b^2)^2-2a^2b^2=29^2-2(100)=641\).
Question · MCQ
Which is a complete factorisation over the integers of \(x^4+4\)?
Correct option and reasoning
Correct option: A. \(x^4+4=(x^2+2)^2-(2x)^2\); use difference of squares to get \((x^2-2x+2)(x^2+2x+2)\).
Question · MCQ
For which values is \((x^2+5x+6)/(x^2-x-6)\) defined, and what is its simplified form?
Correct option and reasoning
Correct option: A. Factor before cancelling: \((x+2)(x+3)/[(x+2)(x-3)]\). The original denominator excludes \(x=-2\) and \(x=3\); cancellation does not restore \(x=-2\).
Question · MCQ
Expand \((x+1)(x+2)(x+3)\).
Correct option and reasoning
Correct option: A. Pair \((x+1)(x+3)=x^2+4x+3\), then multiply by \((x+2)\): \(x^3+6x^2+11x+6\).
Question · MCQ
Factorise completely over the integers: \(x^6-64\).
Correct option and reasoning
Correct option: A. Apply difference of cubes: \((x^2)^3-4^3=(x^2-4)(x^4+4x^2+16)\). Then factor \(x^2-4=(x-2)(x+2)\). Finally \(x^4+4x^2+16=(x^2-2x+4)(x^2+2x+4)\).
Question · MCQ
If \(a+b=10\) and \((a-b)^2=16\), find \(ab\).
Correct option and reasoning
Correct option: A. \((a-b)^2=(a+b)^2-4ab\); hence \(16=100-4ab\), so \(ab=21\).
Question · MCQ
Evaluate \(999^3+1\).
Correct option and reasoning
Correct option: A. \(999^3+1^3=(999+1)(999^2-999+1)=1000(997003)=997003000\).
Question · MCQ
Simplify \((x+2y)^3+(x-2y)^3\).
Correct option and reasoning
Correct option: A. Add the two cube expansions: the terms odd in \(y\) cancel, leaving \(2x^3+6x(2y)^2=2x^3+24xy^2\).
Question · MCQ
Factorise \(x^4+4x^2+16\).
Correct option and reasoning
Correct option: A. Recognise a difference of squares: \((x^2+4)^2-(2x)^2=(x^2-2x+4)(x^2+2x+4)\).
Question · MCQ
Solve over the reals: \(x^4-10x^2+9=0\).
Correct option and reasoning
Correct option: A. Set \(t=x^2\); then \(t^2-10t+9=(t-1)(t-9)=0\). Thus \(x^2=1\) or \(x^2=9\), giving all four real roots.
Question · MCQ
Given \(a+b+c=6\), \(ab+bc+ca=11\) and \(abc=6\), find \((a+b)(b+c)(c+a)\).
Correct option and reasoning
Correct option: A. Use \((a+b)(b+c)(c+a)=(a+b+c)(ab+bc+ca)-abc\); the value is \(6(11)-6=60\).
Question · MCQ
If \(x^2+1/x^2=7\), find \(x^4+1/x^4\).
Correct option and reasoning
Correct option: A. Square the given sum: \((x^2+1/x^2)^2=x^4+2+1/x^4=49\); subtract 2 to get \(47\).