Why do you sometimes finish the square (ex.) and sometimes you make one side zero(ex)? How should I know the difference? Any help is appreciated! TIA

Lets say there's a quadratic equation → ax² + bx = -c

Lets make one side zero which gives → ax² + bx + c = 0

From the above equation ,

For example :- x² - 12x = -35.

Lets make one side zero → x² - 12x + 35 = 0 . Now, lets break -12x into two real numbers 'p' & 'q' such that p + q = -12x & pq = 35. Prime factors of 35 are 1 , 5 & 7. If we take 5 & 7 as 'p' & 'q' then p + q = 5 + 7 = 12. but as we need -12 , the values of 'p' & 'q' are -5 & -7.

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Lets take another example :- x² + 4x = 1.

Lets make one side zero → x² + 4x - 1 = 0 . Now , lets split +4x into into two real numbers 'p' & 'q' such that p + q = +4x & pq = -1 . Prime factors of 4 are 1 & 2 only. But there's no such combination of its primes such that their sum would be equal to 4. So, the quadratic equation can't be solved by factorisation method. Now, lets bring -1 to Right Hand Side & try solving by completing squares method.

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Conclusion:

First try making one side zero and try to solve the equation by factorisation method. If it got factorised , then you are done and if it didn't get factorised , then solve it by completing squares method.

I'm assuming we're asking about the difference between:

and

Let's preface this by stating the obvious: there's no actual difference. Both equations are equivalent.

Now, it has a lot to do with our goal. Sometimes our goal is to write an equation concisely (then the "no-zero" way uses less characters), and sometimes we're dealing with a quadratic function of

in the second case, we wouldn't write

because it'd be confusing. That means that when we're looking for y=0 points of a quadratic equation, the way to convey that goal is that we're looking for the point where - so we preserve the structure of the quadratic function.

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step-by-step explanation: