Algebra - Introduction
In algebra we use symbols to represent terms in a mathematical expression.
For example we can use an algebraic expression for the kinetic energy of a moving object (MV2), or the velocity of a body experiencing acceleration due to gravity (ut + ½ at2).
Most often we relate the value of two arbitrary variables such as X and Y by using equations. These generally take the form of either linear or quadratic equations which would be expressed on a graph as straight lines or curves (parabola) respectively.
Common tasks with worked examples
At GCSE level, it is useful to be able to perform a specific variety of tasks with algebraic expressions and equations. These are summarised below with worked examples.
1.) Simplify - gather together ‘like’ terms x2 + 5x + 6 + 3x2 - 7x - 3 - 6x - 2x2=> (x2 + 3x2 - 2x2) + (5x - 7x -6x) + (6 - 3)
=> (2x2) + (-8x) + (3)
=> 2x2 - 8x + 3
2.) Substitute - put a value into an expression so that it may be evaluated numerically
y = 15x - 5, find the value of y when x is 3y = (15*3) - 5 substitute in the value for x
y = 45 - 5 = 40
3.) Rearrange - ‘Change the subject’ of an equation
y = 3x2 - 5 rearrange to give an expression for x
y + 5 = 3x2 add 5 to both sides of the equation
(y + 5)/3 = x2 divide by 3
x = SQRT((y+5 )/ 3) evaluate square root
4.) Solve - rearrange an equation so that the value of the unknown may be quantified
15 = 5x2 - 5
=> 20 = 5x2
=> 4 = x2
=> x = 2
5.) Expand - multiply out brackets
(x + 5)(x-2)
= x(x-2) + 5(x-2)
= x2 - 2x + 5x - 10
= x2 +3x - 10
6.) Factorise - reverse the above process
X2 + 7x + 6 factorise this expression=> (x + __ ) (x + __ ) the two numbers must multiply to give 6 and add to give 7
try (x + 3) (x + 2) this does not work out
try (x + 6)(x + 1) this is correct
Practice questions
1. Simplify:
(a) 3x2 + 6x - x2 + 5 - 12 + x2
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(b) 6x2 + 3x - 5x2 + 7x - 1 + x2
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(c) 7x2 - 3x - 2x2 + 50 - 10x + 3x2 ______________________________________________________________________
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(d) x2 + 16x - 2x + 5x - 2 + x2
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(e) 4x2 - 3x - 2x2 + 5 - 12x - x2
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(f) 3x2 + 6x - x2 + 5 - 12 + x2
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2. Substitute in the values 5 and -3 for x, and evaluate y in each case:
(a) y = 7x - 5
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(b) y = 6x + 1
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(c) y = 5x2 + x + 1
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(d) y = 16x2 - 7x - 12
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(e) y = 2x3 + 3x2 + 17x - 1
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(f) y = x
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3. Rearrange the following to give an expression for x in terms of y:
(a) y = 3x2 - 5
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(b) y = 6x - 5
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(c) y = (x - 5)(x + 5)
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(d) y = 35x2 - 200
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(e) y = x3 + 12
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(f) y = 12 + 15 sin x
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4. Solve to find the value of x:
(a) x + 5 = 20
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(b) x - 5 = 3x + 1
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(c) 28 = x2 + 3
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(d) x2 +12 = 48
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(e) (x + 1 )(x - 1) = 0
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(f) x2 + 5x - 6 = 0
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5. Expand the following by multiplying out the brackets:
(a) (x + 1)(x - 1)
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(b) (x + 5) (x - 5)
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(c) (x - 7)(x + 8)
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(d) (3x + 1)(2x -1)
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(e) (4x - 1)(6x - 54)
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(f) (x - 1)(x - 1)
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6. Factorise the following expressions :
(a) x2 - 1
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(b) x2 - 25
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(c) x2 - 7x + 6
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(d) x2 - x - 6
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(e) x2 + 5x - 6
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(f) x2 + x - 6
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