EXPANSION OF QUADRATIC EXPRESSIONS

In Secondary 1, we have learnt about the expansion and simplification of linear expressions. The expression that we got have the highest power of 1 for linear expressions. For expansion, we would have to ensure that there are no more brackets in the equation.
An example of a linear equation which needs to be expanded would be
2a(x-y+2c) =2ax-2ay+4ac
For quadratic expression, the highest power is 2.
Similarly, to expand the expression, we would have to expand the expression like we did for the linear expression. One example of expansion of a quadratic expression would be (x+2)(x+4)
The solving:
(x+2)(x+4)
(taken from Shinglee Math TB)
We would have to do it step-by-step. Firstly, we would have to take x to expand both the variables in the next bracket including their signs. We would then have to take x and multiply it by the x and +4. The x would now become (x×x) x2 and the 4 would become (x×4) 4x.
(x+2)(x+4)
Next, take the +2 and again multiply it to the variables in the bracket beside it such that x would now become (2×x) 2x and 4 would become (2×4) 8.
We would then have to add the answers obtained
X2+4x+2x+8 = x2+6x+8
Thus, the expanded form of (x+2) (x+4) is x2+6x+8.

Another example of expansion of quadratic expression would be
(2x+3) (5x-2)-2(5x-3) (x+1)
(taken from Shinglee Math TB)
To solve the expression, we would have to do it step-by-step.
(2x+3) (5x-2)-2(5x-3) (x+1)
= (10x2-4x+15x-6)-2(5x-3) (x+1)
= (10x2-4x+15x-6)-2(5x2+5x-3x-3)
= 10x2 +11x-6-10x2-10x+6x+6
= 7x
Eventually, we would arrive at the answer 7x.

Another example on expansion would be (taken from Shinglee Math TB)
(4x-3)(x+2)-(3x-5)(-x-9)
= (4x2+8x-3x-6) - (3x-5)(-x-9)
= (4x2+8x-3x-6) –(-3x2-27x+5x+45)
= 4x2+8x-3x-6+3x2+27x-5x-45
= 7x2+27x-51
There is also expansion using the special identity.
 (a+b) 2 = a2+2ab+b2
 (a-b) 2 = a2-2ab+b2
 a2-b2 = (a+b)(a-b)

Sometimes we would have some expression that can be factorised by using the special identity. A few questions would be as such:
For the first identity, a possible question would be
 (a+4)2
a2+2(a)(4)+42
= a2+8a+16
For the second identity, a possible question would be:
 (18b-4g)2
= (18b)2-2(18b)(4g)+(4g)2
= 324b2-144bg+16g2
Lastly, for the last identity, a possible question could look like:
 64g2-16a2
= (8g+4a)(8g-4a)

Summary for expansion:
When we expand an expression, we are ensuring that no more brackets are in the expression. Expansion enables us to see and simplify our answers. It means that we have to multiply the variables out of the brackets. When a quadratic expression is expanded, the highest power would be 2.