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Math test - complete the square method

Erm, I took the maths test on thursday, got one question supposed to use complete the square to prove that one value is the least, and thus all values obtained by that equation is positive.

Since this topic is tested under the miscellaneous section of EOY math exam... so i decided to surf wikipedia and blog abt completing the square method.

I will give an example, try understand on ur own first before looking down at my explainations...

$\begin{align}5x^2 + 7x - 6 &{}= 5\left(x^2 + {7 \over 5}x\right) -6 \\ &{}= 5\left(x^2 + {7 \over 5}x +\left({7 \over 10}\right)^2\right) - 6 - 5\left({7 \over 10}\right)^2 \\ &{}= 5\left(x + {7 \over 10}\right)^2 - 6 - {7^2 \over 2\cdot 10} \\ &{}= 5\left(x + {7 \over 10}\right)^2 - {6\cdot 20 + 7^2 \over 20} \\ &{}= 5\left(x + {7 \over 10}\right)^2 - {169 \over 20}. \end{align}$

err, first u sub out the A (A means the constant infront of x with a power 2, which is 5) and u end up with x² (the constant infront of x² must be 1) then the x¹ component will be divided by wadever u factorised out (which is 5 in this case and u end up with 7x/5) .

Now u must do smthing that is impt yet very stupid... u add something and minus something. Recalling wad you noe abt a perfect square quardratic equation: a² + 2ab + b², the third component, b² can be obtained by observing the second component (in this case its 7x/5) and u divide the component by 2 (which means denominator times 2) and square the whole thing.

Not forgetting the minused factor... which u haf to move out and join the constant. Dun forget to times the minused factor by wad u factorised (which is 5) and then u are almost close to finishing.

After dat, just move around the numbers, joining wadever u can join together, u should get smthing like this A(x+B) + C * then u can prove that x is wad then the A(x+B) is equals to 0, thus equation = C but when X is anything bigger/smaller than 'any value', the equation will be bigger than C thus "the value of C" is the smallest.... after obtaining this fact, u can prove other things, like the equation is forever positive (if C is positive) or equation is forever negative (if C is negative).

* note: the signs within can change... depending on the equations given.

dats done, can ask me personally if u are still confused as afterall my english is not dat good...

pictures taken from http://en.wikipedia.org/wiki/Complete_the_square,
credits go to wikipedia, the free encyclopedia!

A sorrow no one can understand nor share..