ABSTRACT
Here are some worked examples to help understanding of brackets with algebra.
Problem 1. Determine 2b(a − 5b)
2b(a − 5b) = 2b× a + 2b×−5b = 2ba − 10b2 = 2ab− 10b2 (note that 2ba is the
same as 2ab)
(3x − y)
(3x + 4y)(x − y) = 3x(x − y)+ 4y(x − y) = 3x2 − 3xy + 4yx − 4y2 = 3x2 − 3xy + 4xy − 4y2 (note that 4yx is the same as 4xy) = 3x2 + xy− 4y2
Problem 3. Simplify 3(2x − 3y)− (3x − y)
3(2x − 3y)− (3x − y) = 3× 2x − 3× 3y − 3x −−y (Note that− (3x − y) = −1(3x − y) and the
−1 multiplies both terms in the bracket) = 6x − 9y − 3x + y
(Note:−×− = +) = 6x − 3x + y − 9y = 3x− 8y
Problem 4. Remove the brackets and simplify the expression (a − 2b)+ 5(b− c)− 3(c+ 2d)
(a − 2b)+ 5(b− c)− 3(c+ 2d) = a − 2b+ 5× b+ 5×−c− 3× c− 3× 2d = a − 2b+ 5b− 5c− 3c− 6d = a+ 3b− 8c− 6d
Problem 5. Simplify (p+ q)(p− q)
(p + q)(p− q) = p(p− q)+ q(p− q) = p2 − pq + qp− q2 = p2 − q2
Problem 6. Simplify (2x − 3y)2
(2x − 3y)2 = (2x − 3y)(2x − 3y) = 2x(2x − 3y)− 3y(2x − 3y) = 2x × 2x + 2x ×−3y − 3y × 2x
−3y ×−3y = 4x2 − 6xy − 6xy + 9y2
(Note: +×− = − and −×− = +) = 4x2 − 12xy+ 9y2
Problem from expression and simplify 2[x2 − 3x(y + x)+ 4xy]
2[x2 − 3x(y + x)+ 4xy]= 2[x2 − 3xy − 3x2 + 4xy] (Whenever more than one type of bracket is involved, always start with the inner brackets)
= 2[−2x2 + xy] = −4x2 + 2xy = 2xy− 4x2
Problem 8. Remove the brackets and simplify the expression 2a − [3{2(4a− b)− 5(a+ 2b)}+ 4a]
(i) Removing the innermost brackets gives 2a − [3{8a− 2b− 5a − 10b}+ 4a]
(ii) Collecting together similar terms gives 2a − [3{3a− 12b}+ 4a]
(iii) Removing the ‘curly’ brackets gives 2a − [9a − 36b+ 4a]
(iv) Collecting together similar terms gives 2a − [13a − 36b]
(v) Removing the outer brackets gives 2a − 13a + 36b
(vi) i.e. −11a+ 36b or 36b− 11a
Now try the following Practice Exercise
Practice Exercise 41 Brackets (answers on page 1112)
Expand the brackets in Problems 1 to 28.
