What is the complete factorisation of (12x^2-27y^2)?
Correct answer: A
First take (3) common and split (4x^2-9y^2) by difference of squares. Exam tip: fully factorise the final answer.
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Factorisation
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Hard · Level 69 · common factor,difference of squares,complete factorisation
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- (3(2x-3y)(2x+3y))
- (3(4x-9y)(x+y))
- (12(x-3y)(x+3y))
- (3(4x^2-9y^2))
Hard · Level 69 · factorisation, algebraic identities, perfect square trinomial, binomial square, class 9 mathematics
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- \(9p^2+12pq+4q^2\)
- \(9p^2+12pq+5q^2\)
- \(9p^2-12pq-4q^2\)
- \(6p^2+12pq+6q^2\)
Hard · Level 69 · higher powers,complete factorisation,difference of squares
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- ((x^2-9)(x^2+9))
- ((x-9)(x+9)(x^2+9))
- ((x-3)(x+3)(x^2+9))
- ((x-3)^2(x+3)^2)
Hard · Level 69 · factorisation,algebraic identities,perfect square,difference of squares,class 9 mathematics
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- \((x-y-5)(x-y+5)\)
- \((x+y-5)(x+y+5)\)
- \((x-y)^2+25\)
- \((x-5)(y+5)\)
Hard · Level 69 · algebraic identities, factorisation, perfect square trinomial, square area, class 9 mathematics
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- \(7a-6b\)
- \(7a+6b\)
- \(49a-36b\)
- \(7a^2-6b^2\)
Hard · Level 69 · factorisation, algebraic identities, perfect square trinomial, class 9 mathematics, quadratic expressions
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- \(9p^2-24pq+16q^2\)
- \(9p^2-20pq+16q^2\)
- \(9p^2+24pq+8q^2\)
- \(9p^2-24pq-16q^2\)
Hard · Level 69 · signed factorisation,quadratic,hard
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- ((3x+4)(2x-3))
- ((6x+3)(x-4))
- ((2x+3)(3x-4))
- ((3x-4)(2x+3))
Hard · Level 69 · factorisation, algebraic identities, difference of squares, class 9 mathematics
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- \(x^2-16\)
- \(9a^2-b^2\)
- \(p^4-q^4\)
- \(4m^2+25n^2\)
Hard · Level 69 · complete factorisation,cubic expression,common factor
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- (x(x+3)(x+2))
- (x(x+1)(x+2))
- (x(x+1)(x+3))
- (x^2(x+3))
Hard · Level 69 · algebraic identities, factorisation, perfect square trinomial, binomial square, class 9 mathematics
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- \(9p^2-24pq+16q^2\)
- \(9p^2-24pq+12q^2\)
- \(9p^2-18pq+16q^2\)
- \(9p^2+20pq+16q^2\)
Hard · Level 69 · factorisation,algebraic identities,common factor,difference of squares,cubic polynomial
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- \(x(x-9)\)
- \(x(x-3)^2\)
- \(x(x-3)(x+3)\)
- \((x-3)(x+3)\)
Hard · Level 69 · factorisation, algebraic identities, difference of squares, perfect squares, class 9 mathematics
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- \(p^2-10p+25-49q^2\)
- \(p^2-10p+25+49q^2\)
- \(p^2+10p+25-49q\)
- \(p^2-25-49q^2\)
Hard · Level 69 · algebraic identities, factorisation, perfect square trinomial, class 9 mathematics, binomial square
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- \(16x^2-24xy+8y^2\)
- \(16x^2+24xy+9y^2\)
- \(16x^2-24xy+9y^2\)
- \(16x^2-12xy+9y^2\)
Hard · Level 69 · algebraic identities, factorisation, perfect cube, binomial cube, class 9 mathematics
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- \(8x^3-36x^2y+54xy^2-27y^3\)
- \(8x^3+36x^2y+54xy^2-27y^3\)
- \(8x^3-36x^2y+54xy^2+27y^3\)
- \(8x^3-18x^2y+27xy^2-27y^3\)
Hard · Level 69 · factorisation, grouping method, algebraic identities, common factor, sign convention
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- \((a+d)(b-c)\)
- \((a-d)(b-c)\)
- \((a+d)(b+c)\)
- \((a-d)(b+c)\)
Hard · Level 69 · algebraic identities, factorisation, quadratic trinomials, area model, class 9 mathematics
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- \((3x+2)\) और \((2x+5)\)
- \((3x+5)\) और \((2x+2)\)
- \((6x+5)\) और \((x+2)\)
- \((3x+10)\) और \((2x+1)\)
Hard · Level 69 · factorisation, algebraic identities, difference of squares, class 9 mathematics, perfect square trinomials
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- \(25p^2-49q^2\)
- \(25p^2+49q^2\)
- \(25p^2-70pq+49q^2\)
- \(25p^2+70pq+49q^2\)
Hard · Level 69 · factorisation, algebraic identities, difference of squares, conjugate binomials, class 9 mathematics
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- \(49x^2-(3y+2)^2\)
- \(49x^2+14x(3y+2)+(3y+2)^2\)
- \(49x^2+(3y+2)^2\)
- \(49x^2-14x(3y+2)+(3y+2)^2\)
Hard · Level 69 · factorisation, algebraic identities, difference of cubes, class 9 mathematics, polynomial factors
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- \(x^3-8y^3\)
- \(x^3+8y^3\)
- \(x^2+4y^2\)
- \(x^2-2xy+4y^2\)
Hard · Level 69 · factorisation, quadratic expressions, algebraic identities, discriminant, integer factors, class 9 mathematics
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- \(x^2+5x+6\)
- \(x^2-9\)
- \(x^2+2x+1\)
- \(x^2+x-1\)
Which of the following trinomials can be factorised as a perfect square of a binomial?
Correct answer: A
\((3p+2q)^2=9p^2+2\cdot3p\cdot2q+4q^2=9p^2+12pq+4q^2\), so A is a perfect square. In B, the last term should be \(4q^2\), not \(5q^2\). Exam tip: compare the middle term with \(2ab\).
View question detailsWhat will be the complete factorisation of (x^4-81)?
Correct answer: C
(x^4-81=(x^2-9)(x^2+9)) and (x^2-9=(x-3)(x+3)). Exam tip: check difference of squares repeatedly.
View question detailsChoose the correct factorisation of (x^2-2xy+y^2-25).
Correct answer: A
The first three terms satisfy \(x^2-2xy+y^2=(x-y)^2\). Hence the expression is \((x-y)^2-25=(x-y)^2-5^2\). Applying \(a^2-b^2=(a-b)(a+b)\) gives \((x-y-5)(x-y+5)\). Option B would involve \((x+y)^2\), whose middle term is \(+2xy\), whereas the given expression has \(-2xy\). Exam tip: identify the perfect-square trinomial before applying the difference-of-squares identity.
View question detailsThe area of a square courtyard is \(49a^2-84ab+36b^2\) square units. If \(7a>6b\), what will be the length of its side?
Correct answer: A
This matches \(u^2-2uv+v^2\). With \(u=7a\) and \(v=6b\), the middle term is \(-2uv=-84ab\), so the area is \((7a-6b)^2\). Since \(7a>6b\), the positive side is \(7a-6b\). In exams, verify the middle term first.
View question detailsWhich of the following trinomials is a perfect-square expression?
Correct answer: A
We have \(9p^2=(3p)^2\) and \(16q^2=(4q)^2\). For a perfect-square trinomial, the middle term must be \(\pm2ab\). Here, \(2\times3p\times4q=24pq\), and the middle term is \(-24pq\). Therefore, \(9p^2-24pq+16q^2=(3p-4q)^2\), so option A is correct. In option B, the middle term is \(-20pq\), not the required \(-24pq\). Exam tip: take the square roots of the first and last terms, then check twice their product against the middle term.
View question detailsChoose the factorisation of (6x^2-x-12).
Correct answer: D
((3x-4)(2x+3)=6x^2+9x-8x-12). The middle term becomes (x), so signs must be checked carefully.
View question detailsWhich of the following expressions cannot be factorised using the identity for the difference of two squares?
Correct answer: D
\(4m^2+25n^2=(2m)^2+(5n)^2\) is a sum of squares, not a difference. The identity is \(a^2-b^2=(a-b)(a+b)\). In exams, check the sign between the two square terms first.
View question detailsWhat is the complete factorisation of (x^3+3x^2+2x)?
Correct answer: B
First take (x) common and factor (x^2+3x+2) as ((x+1)(x+2)). Exam tip: factor the trinomial after common factor.
View question detailsWhich of the following expressions is the square of the binomial \(3p-4q\)?
Correct answer: A
Use \((a-b)^2=a^2-2ab+b^2\). With \(a=3p\) and \(b=4q\), the middle term is \(-2\times3p\times4q=-24pq\), and the last term is \(16q^2\). Hence A is correct. Exam tip: always check the middle-term sign.
View question detailsWhat is the complete factorisation of (x^3-9x)?
Correct answer: C
In \(x^3-9x\), \(x\) is the common factor, so \(x^3-9x=x(x^2-9)\). Now \(x^2-9=x^2-3^2\) is a difference of squares; hence \(x^2-9=(x-3)(x+3)\). Therefore, the complete factorisation is \(x(x-3)(x+3)\). Option B is incorrect because \((x-3)^2\) would introduce a \(-6x^2\) term on expansion. Exam tip: take out the common factor first, then look for a difference of squares.
View question detailsWhich of the following expressions can be factorised using the identity \((A^2-B^2)=(A-B)(A+B)\) without regrouping its terms?
Correct answer: A
In option A, \(p^2-10p+25=(p-5)^2\) and \(49q^2=(7q)^2\), so it is a difference of two squares. Option B is a sum of squares. Exam tip: identify perfect squares first.
View question detailsWhich of the following expressions is a perfect square of a binomial based on the identity \(a^2-2ab+b^2=(a-b)^2\)?
Correct answer: C
Here \(16x^2=(4x)^2\) and \(9y^2=(3y)^2\). The middle term is \(-24xy=-2(4x)(3y)\), so the expression is \((4x-3y)^2\). In exams, verify the middle term against \(\pm2ab\).
View question detailsWhich of the following expressions is a perfect cube of a binomial and can therefore be factorised using a cube identity?
Correct answer: A
Using \((a-b)^3=a^3-3a^2b+3ab^2-b^3\), put \(a=2x\) and \(b=3y\). This gives option A, \((2x-3y)^3\). Option C has the wrong sign for the last term. Exam tip: check the first and last terms first.
View question detailsWhich is the factorisation of (ab-ac+db-dc)?
Correct answer: A
Group the terms as \(ab-ac+db-dc=a(b-c)+d(b-c)\). Since \((b-c)\) is common to both groups, taking it out gives \((a+d)(b-c)\). In option C, \((b+c)\) would make the \(ac\) and \(dc\) terms positive, so it is incorrect. Exam tip: Check the signs of every term carefully while grouping.
View question detailsThe area of a rectangular garden is \(6x^2+19x+10\) square units. Which of the following pairs can represent its possible length and breadth?
Correct answer: A
Split the middle term as \(15x+4x\): \(3x(2x+5)+2(2x+5)=(3x+2)(2x+5)\). Hence A is correct. Option B gives a middle term of \(16x\). Exam tip: multiply the factors to verify the middle term.
View question detailsWhich of the following expressions can be factorised directly using the identity for the difference of squares, \(a^2-b^2=(a-b)(a+b)\)?
Correct answer: A
\(25p^2-49q^2=(5p)^2-(7q)^2\), so it is a difference of squares and factors as \((5p-7q)(5p+7q)\). B has a sum, while C is a perfect square. Exam tip: check for two squares joined by a minus sign.
View question detailsWhich of the following expressions can be factorised as a product of conjugate binomials?
Correct answer: A
In option A, \(49x^2=(7x)^2\), so it has the form \((7x)^2-(3y+2)^2\), a difference of squares. Hence its factors are \((7x-3y-2)(7x+3y+2)\). Exam tip: first identify whether both terms are perfect squares with a minus sign.
View question detailsWhich of the following expressions has \(x-2y\) as a factor in its complete factorisation?
Correct answer: A
\(x^3-8y^3=x^3-(2y)^3=(x-2y)(x^2+2xy+4y^2)\), so \(x-2y\) is a factor. In \(x^3+8y^3\), the factor is \(x+2y\). Exam tip: identify the difference-of-cubes pattern first.
View question detailsWhich of the following quadratic expressions cannot be factorised into two linear factors with integer coefficients?
Correct answer: D
For \(x^2+x-1\), the discriminant is \(b^2-4ac=1+4=5\), which is not a perfect square. Hence it has no linear factors with integer coefficients. In A, 2 and 3 multiply to 6 and add to 5. Exam tip: check the discriminant first.
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