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→Factoring
This is just shorthand for "x is either 2 or 3."
This is just shorthand for "x is either 2 or 3."
===Completing the square===
Completing the square is a method of solving quadratic equations that involves turning the left-hand side of the equation into a perfect square trinomial and then taking the square root of both sides. In our original equation:
::<math>x^2 - 5x + 6 = 0\,</math>
the left-hand side is not a perfect square trinomial. That is to say, if we tried to take the square root of it, we couldn't simplify it any further, so that wouldn't help us in solving the equation. We need to change the equation to make it a perfect square.
The key here is the coefficient of the linear term, which is the number that's multiplied by the x that isn't squared. For us, that's 5. We divide that number by 2 and then square it, which yields
::<math>(5/2)^2 = 25/4 = 6 + 1/4\,</math>
So if we add 1/4 to both sides, the left side will be a perfect square trinomial. Thus our new equation is:
::<math>x^2 - 5x + 6 + 1/4 = 1/4\,</math>
Or, equivalently:
::<math>(x - 5/2)^2 = 1/4\,</math>
We can now take the square root:
::<math>x - 5/2 = +- 1/2\,</math>
The "+-" indicates that the 1/2 is either positive or negative. The result is thus
::<math>x = 5/2 +- 1/2\,</math>
Since 5/2 + 1/2 = 3, and 5/2 - 1/2 = 2, this is equivalent to the result obtained through factoring.
==Notes and references==
==Notes and references==