COMPLETING THE SQUARE (PENYEMPURNAAN KUASA DUA)
I. Basic Ideas
There are two types of factorised quadratic equations, eg:
i) x2 + 6x + 8 = (x + 2)(x + 4)
ii) x2 + 6x + 9 = (x + 3)2
In type (i), the factors are different. In type (ii), the factors are the same:
x2 + 6x + 9 = (x + 3)(x + 3)
Completing the square (Penyempurnaan kuasa dua) is about writing the equation in a way so that you can “put a square (kuasa-dua) on a bracket” like the (x + 3)2. (That means converting the equation from y = ax2 + bx + c into y = a(x - p)2 + q. Note the + and - signs... DO NOT get them mixed up!)
Now, it is possible to do the same with equation (i) up there. You just have to change the 8:
x2 + 6x + 8 = x2 + 6x + 9 – 1
= (x + 3)2 – 1
As you can see, I made 8 = (9 – 1). You should be able to see why -- Compare all the blue and red sections with equation (ii) above.
Similarly,
x2 + 6x + 12 = x2 + 6x + 9 + 3
= (x + 3)2 + 3
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