Hard GCSE Simplifying and single brackets Questions

Challenging, exam-style GCSE Simplifying and single brackets questions with worked solutions. Stretch yourself on the hardest expanding brackets, collecting like terms, negative multiplier, factorising problems.

expanding bracketscollecting like termsnegative multiplierfactorisingvariable factordividing terms
GCSE Foundation34 questionsStep-by-step solutions
Question 1
6 markschallenging
Expand and simplify 3x(x+4)+2(x2−x)−5x3x(x + 4) + 2(x^2 - x) - 5x.
Show worked solution

Worked solution

  1. Expand the first bracket

    3x(x+4)=3x2+12x3x(x + 4) = 3x^2 + 12x

    Multiply x and 4 by 3x.

  2. Expand the second bracket

    2(x2−x)=2x2−2x2(x^2 - x) = 2x^2 - 2x

    Multiply x2x^2 and x by 2.

  3. Bring down the last term

    −5x- 5x

    The lone term is −5x-5x.

  4. Group the x2x^2 terms

    3x2+2x2=5x23x^2 + 2x^2 = 5x^2

    3+2=53 + 2 = 5.

  5. Group the x terms

    12x−2x−5x=5x12x - 2x - 5x = 5x

    12−2−5=512 - 2 - 5 = 5.

  6. Write the result

    5x2+5x5x^2 + 5x

    Put the x2x^2 term first, then the x term.

  7. Check by substitution

    x=2:original=30,answer=30x = 2:\quad \text{original} = 30,\quad \text{answer} = 30

    Substituting x=2x = 2 into the original expression and into the answer both give 3030, so the simplification is correct.

  8. Note the general rule

    a(b+c)=ab+aca(b + c) = ab + ac

    To expand a single bracket, multiply the term outside by every term inside.

  9. Check by substitution

    x=3:original=60,answer=60x = 3:\quad \text{original} = 60,\quad \text{answer} = 60

    Substituting x=3x = 3 into the original expression and into the answer both give 6060, so the simplification is correct.

  10. Avoid the common error

    3(2x−5)≠6x−53(2x - 5) \ne 6x - 5

    A common mistake is to multiply only the first term. Every term inside the bracket must be multiplied.

  11. Check by substitution

    x=1:original=10,answer=10x = 1:\quad \text{original} = 10,\quad \text{answer} = 10

    Substituting x=1x = 1 into the original expression and into the answer both give 1010, so the simplification is correct.

  12. Reflect

    ✓\checkmark

    The expression is now fully simplified and cannot be reduced any further.

  13. Check by substitution

    x=5:original=150,answer=150x = 5:\quad \text{original} = 150,\quad \text{answer} = 150

    Substituting x=5x = 5 into the original expression and into the answer both give 150150, so the simplification is correct.

  14. Check by substitution

    x=10:original=550,answer=550x = 10:\quad \text{original} = 550,\quad \text{answer} = 550

    Substituting x=10x = 10 into the original expression and into the answer both give 550550, so the simplification is correct.

  15. State the answer

    5x2+5x5x^2 + 5x

    The simplified expression is 5x2+5x5x^2 + 5x.

Answer
5x2+5x5x^2 + 5x
Question 2
6 markschallenging
Expand and simplify 6(x+2)−2(2x−3)+46(x + 2) - 2(2x - 3) + 4.
Show worked solution

Worked solution

  1. Expand the first bracket

    6(x+2)=6x+126(x + 2) = 6x + 12

    Multiply x and 2 by 6.

  2. Expand the second bracket

    −2(2x−3)=−4x+6-2(2x - 3) = -4x + 6

    A negative times a negative gives +66.

  3. Bring down the constant

    +4+ 4

    The lone 44 has no bracket.

  4. Group the x terms

    6x−4x=2x6x - 4x = 2x

    6−4=26 - 4 = 2.

  5. Group the numbers

    12+6+4=2212 + 6 + 4 = 22

    Add the constants.

  6. Write the result

    2x+222x + 22

    Combine the x term and the constant.

  7. Check by substitution

    x=2:original=26,answer=26x = 2:\quad \text{original} = 26,\quad \text{answer} = 26

    Substituting x=2x = 2 into the original expression and into the answer both give 2626, so the simplification is correct.

  8. Note the general rule

    a(b+c)=ab+aca(b + c) = ab + ac

    To expand a single bracket, multiply the term outside by every term inside.

  9. Check by substitution

    x=3:original=28,answer=28x = 3:\quad \text{original} = 28,\quad \text{answer} = 28

    Substituting x=3x = 3 into the original expression and into the answer both give 2828, so the simplification is correct.

  10. Avoid the common error

    3(2x−5)≠6x−53(2x - 5) \ne 6x - 5

    A common mistake is to multiply only the first term. Every term inside the bracket must be multiplied.

  11. Check by substitution

    x=1:original=24,answer=24x = 1:\quad \text{original} = 24,\quad \text{answer} = 24

    Substituting x=1x = 1 into the original expression and into the answer both give 2424, so the simplification is correct.

  12. Reflect

    ✓\checkmark

    The expression is now fully simplified and cannot be reduced any further.

  13. Check by substitution

    x=5:original=32,answer=32x = 5:\quad \text{original} = 32,\quad \text{answer} = 32

    Substituting x=5x = 5 into the original expression and into the answer both give 3232, so the simplification is correct.

  14. Check by substitution

    x=10:original=42,answer=42x = 10:\quad \text{original} = 42,\quad \text{answer} = 42

    Substituting x=10x = 10 into the original expression and into the answer both give 4242, so the simplification is correct.

  15. State the answer

    2x+222x + 22

    The simplified expression is 2x+222x + 22.

Answer
2x+222x + 22
Question 3
5 markschallenging
A rectangle has length (3x+2)(3x + 2) cm and width (x−1)(x - 1) cm. Write and simplify an expression for its perimeter.
Show worked solution

Worked solution

  1. Write the perimeter

    P=2((3x+2)+(x−1))P = 2\big((3x + 2) + (x - 1)\big)

    Perimeter is twice the length plus the width.

  2. Add length and width

    (3x+2)+(x−1)=4x+1(3x + 2) + (x - 1) = 4x + 1

    Collect like terms inside the bracket.

  3. Double the result

    2(4x+1)2(4x + 1)

    Multiply the whole bracket by 22.

  4. Expand

    2×4x+2×12 \times 4x + 2 \times 1

    Multiply each term by 22.

  5. Work out each product

    8x+28x + 2

    22 times 4x4x is 8x8x and 22 times 11 is 22.

  6. Interpret the answer

    P=8x+2P = 8x + 2

    This is the perimeter in centimetres.

  7. Check by substitution

    x=2:original=18,answer=18x = 2:\quad \text{original} = 18,\quad \text{answer} = 18

    Substituting x=2x = 2 into the original expression and into the answer both give 1818, so the simplification is correct.

  8. Note the general rule

    P=sum of all sidesP = \text{sum of all sides}

    Add the side expressions, then collect like terms.

  9. Check by substitution

    x=3:original=26,answer=26x = 3:\quad \text{original} = 26,\quad \text{answer} = 26

    Substituting x=3x = 3 into the original expression and into the answer both give 2626, so the simplification is correct.

  10. Avoid the common error

    P≠length×widthP \ne \text{length} \times \text{width}

    Perimeter is the total distance around the shape, not the area.

  11. Check by substitution

    x=1:original=10,answer=10x = 1:\quad \text{original} = 10,\quad \text{answer} = 10

    Substituting x=1x = 1 into the original expression and into the answer both give 1010, so the simplification is correct.

  12. Reflect

    ✓\checkmark

    The expression is now fully simplified and cannot be reduced any further.

  13. Check by substitution

    x=5:original=42,answer=42x = 5:\quad \text{original} = 42,\quad \text{answer} = 42

    Substituting x=5x = 5 into the original expression and into the answer both give 4242, so the simplification is correct.

  14. Check by substitution

    x=10:original=82,answer=82x = 10:\quad \text{original} = 82,\quad \text{answer} = 82

    Substituting x=10x = 10 into the original expression and into the answer both give 8282, so the simplification is correct.

  15. State the answer

    8x+28x + 2

    The perimeter is (8x+28x + 2) cm.

Answer
8x+28x + 2
Question 4
6 markschallenging
Expand and simplify 4x(2x−1)−3x(x−5)4x(2x - 1) - 3x(x - 5).
Show worked solution

Worked solution

  1. Expand the first bracket

    4x(2x−1)=8x2−4x4x(2x - 1) = 8x^2 - 4x

    Multiply 2x2x and 11 by 4x4x.

  2. Expand the second bracket

    −3x(x−5)=−3x2+15x-3x(x - 5) = -3x^2 + 15x

    A negative times a negative gives +15x15x.

  3. Group the x2x^2 terms

    8x2−3x2=5x28x^2 - 3x^2 = 5x^2

    8−3=58 - 3 = 5.

  4. Group the x terms

    −4x+15x=11x-4x + 15x = 11x

    −4+15=11-4 + 15 = 11.

  5. Write the result

    5x2+11x5x^2 + 11x

    Put the x2x^2 term first, then the x term.

  6. Note the common factor

    5x2+11x=x(5x+11)5x^2 + 11x = x(5x + 11)

    A factor of x can be taken out as a check.

  7. Check by substitution

    x=2:original=42,answer=42x = 2:\quad \text{original} = 42,\quad \text{answer} = 42

    Substituting x=2x = 2 into the original expression and into the answer both give 4242, so the simplification is correct.

  8. Note the general rule

    a(b+c)=ab+aca(b + c) = ab + ac

    To expand a single bracket, multiply the term outside by every term inside.

  9. Check by substitution

    x=3:original=78,answer=78x = 3:\quad \text{original} = 78,\quad \text{answer} = 78

    Substituting x=3x = 3 into the original expression and into the answer both give 7878, so the simplification is correct.

  10. Avoid the common error

    3(2x−5)≠6x−53(2x - 5) \ne 6x - 5

    A common mistake is to multiply only the first term. Every term inside the bracket must be multiplied.

  11. Check by substitution

    x=1:original=16,answer=16x = 1:\quad \text{original} = 16,\quad \text{answer} = 16

    Substituting x=1x = 1 into the original expression and into the answer both give 1616, so the simplification is correct.

  12. Reflect

    ✓\checkmark

    The expression is now fully simplified and cannot be reduced any further.

  13. Check by substitution

    x=5:original=180,answer=180x = 5:\quad \text{original} = 180,\quad \text{answer} = 180

    Substituting x=5x = 5 into the original expression and into the answer both give 180180, so the simplification is correct.

  14. Check by substitution

    x=10:original=610,answer=610x = 10:\quad \text{original} = 610,\quad \text{answer} = 610

    Substituting x=10x = 10 into the original expression and into the answer both give 610610, so the simplification is correct.

  15. State the answer

    5x2+11x5x^2 + 11x

    The simplified expression is 5x2+11x5x^2 + 11x.

Answer
5x2+11x5x^2 + 11x
Question 5
6 markschallenging
Factorise fully 12x2y+8xy12x^2 y + 8xy.
Show worked solution

Worked solution

  1. Find the number factor

    HCF of 12 and 8=4\text{HCF of } 12 \text{ and } 8 = 4

    44 divides both 1212 and 88.

  2. Find the x factor

    x2 and x share xx^2 \text{ and } x \text{ share } x

    Both terms contain at least one x.

  3. Find the y factor

    both terms contain y\text{both terms contain } y

    y appears in each term.

  4. State the full common factor

    4xy4xy

    Combine the number and both letters.

  5. Divide the first term

    12x2y÷4xy=3x12x^2 y \div 4xy = 3x

    Divide numbers and subtract indices.

  6. Divide the second term

    8xy÷4xy=28xy \div 4xy = 2

    Everything cancels except 22.

  7. Check by substitution

    x=2, y=3:original=192,answer=192x = 2,\ y = 3:\quad \text{original} = 192,\quad \text{answer} = 192

    Substituting x=2x = 2, y=3y = 3 into the original expression and into the answer both give 192192, so the simplification is correct.

  8. Note the general rule

    ab+ac=a(b+c)ab + ac = a(b + c)

    To factorise, take out the highest common factor, then write what is left inside the bracket.

  9. Check by substitution

    x=3, y=2:original=264,answer=264x = 3,\ y = 2:\quad \text{original} = 264,\quad \text{answer} = 264

    Substituting x=3x = 3, y=2y = 2 into the original expression and into the answer both give 264264, so the simplification is correct.

  10. Avoid the common error

    6x2+9x≠3(2x2+3x)6x^2 + 9x \ne 3(2x^2 + 3x)

    Taking out only the number is not full factorisation — if the terms still share a letter, include it in the common factor.

  11. Check by substitution

    x=1, y=4:original=80,answer=80x = 1,\ y = 4:\quad \text{original} = 80,\quad \text{answer} = 80

    Substituting x=1x = 1, y=4y = 4 into the original expression and into the answer both give 8080, so the simplification is correct.

  12. Reflect

    ✓\checkmark

    The expression is now fully simplified and cannot be reduced any further.

  13. Check by substitution

    x=4, y=1:original=224,answer=224x = 4,\ y = 1:\quad \text{original} = 224,\quad \text{answer} = 224

    Substituting x=4x = 4, y=1y = 1 into the original expression and into the answer both give 224224, so the simplification is correct.

  14. Check by substitution

    x=2, y=5:original=320,answer=320x = 2,\ y = 5:\quad \text{original} = 320,\quad \text{answer} = 320

    Substituting x=2x = 2, y=5y = 5 into the original expression and into the answer both give 320320, so the simplification is correct.

  15. State the answer

    4xy(3x+2)4xy(3x + 2)

    The fully factorised form is 4xy(3x+23x + 2).

Answer
4xy(3x+2)4xy(3x + 2)

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