Question 4794: A quadratic equation always has two solutions. How do you account for
this difference when compared to linear equations? At what step in the
process do we ensure that we get two solutions when using the
Completing the Square method?
Answer by rapaljer(4671)   (Show Source):
You can put this solution on YOUR website! In any polynomial equation, the degree or order of the polynomial is the highest power of the variable in the equation. The degree of the polynomial gives you the total number of solutions to the equation. For example, a linear equation is a simple "x-to-the-first-power" equation, has one solution. A quadratic equation is of second degree, so it has two solutions. A cubic equation  will have three solutions, etc.
However, having said that, it is possible that a quadratic equation might have two solutions that are actually the same number. For example,  can be factored into  or  . In this case, the TWO solutions are x= 3 and x=3. We say that this is a "double root (solution)" or that the solution x=3 is of "multiplicity 2." Keep in mind that the two solutions in a quadratic equation might be complex solutions, and in this case there are no real solutions at all.
In completing the square process, when you have something like  and you must take the square root of both sides to "undo" the square in order to solve for x, you MUST put the PLUS or MINUS in order to get both solutions. The answer here will be  . Finally add 5 to each side of the equation to get  ,
which is actually TWO solutions. If you forget the PLUS or MINUS, then you automatically lose half of your answers.
R^2 at SCC
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