Alegbra
Algebra is used in Maths when we do not know the exact number(s) in
a calculation.
Solve for x, when x+18 = 26
X=26-18
X= 8
It helps in the illustration of problems or situations as mathematical
expressions. It includes variables like x, y, z, and mathematical
operations like addition, subtraction, multiplication, and division to
form a meaningful mathematical expression. All the branches of
mathematics such as trigonometry, calculus, coordinate geometry,
involve the use of algebra. One simple example of an expression in
algebra is 3x + 2= 14
Algebra deals with symbols and these symbols are related to each
other with the help of operators. It is not just a mathematical concept,
but a skill that all of us use in our daily life without even realizing it.
Understanding algebra as a concept is more important than solving
equations and finding the right answer.
Define key vocabulary and notations used in algebra
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What is an equation ?
In algebra, an equation can be defined as a mathematical statement consisting of an
equal symbol between two algebraic that have the same value.
The most basic and common algebraic equations in math consist of one or more
variables.
For instance, 2x + 6 = 12 is an equation, in which 2x + 6 and 12 are two expressions
separated by an ‘equal’ sign.
In an algebraic equation, the left-hand side is equal to the right-hand side.
Here, for example, 5x + 9 is the expression on the left-hand side, which is equal to
the expression 24 on the right-hand side.
The process of finding the value of the variable is called solving the equation.
• There are different names for different parts
of an equation
Lets say we have an equation 2x +3 =7
2x + 3 =7
where the coefficient is 2
Variable is the symbol for a number we don't
know yet, which is x and consonant is the
number on its own which is 7.
A Term is either a single number or a variable, or
numbers and variables multiplied together.
An Expression is a group of terms (the terms are
separated by + or − signs).
Multiplying and division with algebra
We can multiply two algebraic terms to get a
product, which is also an algebraic term.
Example:
Evaluate 2ab3× 3de
Solution:
2ab3× 3de
=2×a×b×b×b×3×d×e
=2×3×a×b×b×b×d×e
= 6 × a × b3 × e
= 6ab3e
• Step 1: Write the division of the algebraic terms as a fraction.
Step 2: Simplify the coefficient.
Step 3: Cancel variables of the same type in the numerator and
denominator.
How to divide Algebraic Expressions?
Step 1: Factorize the algebraic expressions.
Step 2: Cancel factors in the numerator and denominator where
possible.
= 8ab3 ÷ 4ab
= 8xaxbxbxb
4xaxb
simplify the coefficient and cancel variables of the same type in
numerator and denominator.
= 2b2
Simplification by collecting like terms
• A term is an individual part of an expression and typically appears in one
of three forms:
A number by itself
A letter by itself
A combination of letters and numbers
Like terms have the same combination of letters. To add or subtract terms
with the same letter, we add or subtract the numbers like usual and just put
the letter back on the end.
simplify the following,
4a + 5 +a + 7
We have to group the terms that are similar; all the terms that are just
numbers need grouping together and simplifying, as do the similar algebraic
terms.
This gives,
4a+a =5a
5 + 7 = 12
This gives the final answer to be,
4a +5+a+7 = 5a + 12
Note: Normally the 1 in front of the a is left out but we have included it here
to make it clear the steps being taken.
Demonstrate expanding linear and
quadration
To expand a bracket means to multiply each term in the bracket by the expression
outside the bracket
Let's take the example: 3(x + 2) + 2(x - 1)
Start with the first part: 3(x + 2)
We have to multiply what is outside of the bracket with the individual things inside the
bracket separately and add them together:
3 x (x) + 3 x (2)
Now SIMPLIFY to get: 3x + 6
Next part, we just do the same method:
2(x - 1) =
2 x (x) + 2 x (-1) =
2x - 2
^^^ This is always a good way to lay out your work!
Now add the two parts together:
(3x + 6) + (2x -2)
Putting the bits with 'x's in them next to each other helps:
(3x + 6) + (2x -2) =
3x + 2x + 6 - 2 =
5x + 4
• Quadratic expressions are algebraic expressions where the variable has an exponent of 2.
Expansion is done by removing the brackets from an equation by multiplying each other. This is done by
taking the number outside the bracket to multiply the numbers inside the bracket.
Examples:
a(b + c) = ab + ac
-a(b + c) = -ab - bc
a(b - c) = ab - ac
When you want to expand an equation with two brackets, the terms in the first bracket must be
multiplied by each term in the equation. This may sound confusing, so let me give you some examples!
Example:
(a + b)(c + d) = a(c + d) + b(c + d) = ac + ad + bc + bd
Take note whenever you deal with negative equations. Do remember to simplify the equation as well!
Example:
(x - 2)(x + 3)
= x(x + 3) - 2(x + 3)
= x² + 3x - 2x - 6
= x² + x - 6
Here are some of the formulas for expansion
(a + b)² = a² + 2ab + b²
(a - b)² = a² - 2ab + b²
(a + b)(a - b) = a² - b²
Factorising linear expression
• To factorise an expression means to 'put into
brackets' by taking out common factors. When
factorising, always take the largest factors
possible out of the expression.
a) Factorise: 3x+9.
Notice 3 is both a factor of 3x and 9. We can
factorise the expression by taking the 3 outside
of the brackets.
3x+9=3(x+3).
Factorising quadratic expression
• To factorise a quadratic expression in the form x2 + bx + c we need
double brackets. Factorisation into double brackets is the reverse
process of expanding double brackets.
In this case, the coefficient (number in front) of the x2 term is 1 (a=1).
These are known as monic quadratics.
In order to factorise a quadratic algebraic expression in the form x2 +
bx + c into double brackets:
Write out the factor pairs of the last number (c).
Find a pair of factors that + to give the middle number (b) and ✕ to
give the last number (c).
Write two brackets and put the variable at the start of each one.
Write one factor in the first bracket and the other factor in the second
bracket. The order isn’t important, the signs of the factors are.
• Factorise : x2+4x+3
Write out the factor pairs of the last number (3) in order
X2+4x+3
Factor of 3 : (1,3)
We need a pair of factors that gives 4 when they are added and 3 when they are
multiplied.
1+3=4
1x3 = 3
to multiply two values together to give a positive answer, the signs must be the same
Write two brackets and put the variable at the start of each one (x in this case).
(x )(x )
Write one factor in the first bracket and the other factor in the second bracket. The
order isn’t important, the signs of the factors are.
(x+1)(x+3)
We have now fully factorised the quadratic expression.
We can check the answer by multiplying out the brackets
(x+1)(x+3) = x2+3x+x+3 = x2+4x+3