Solve this quadratic equation by completing the square. x^2 - 3x + 5 = 0

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steveschoen | College Teacher | (Level 1) Associate Educator

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Hi, rosey,


First, we would subtract 5 from each side, giving us:

x^2 - 3x = -5

As well as we have a=1.  So, then, we take half b, then square that.  So, half of 3 is 1.5.  Square 1.5, or 1.5^2.  That is 2.25.  Add that to each side, giving us. . .

x^2 - 3x + 2.25 = -5 + 2.25

x^2 - 3x + 2.25 = -2.75


Then, the left side is factorable after this step always in the form of (x +- #)^2.  For this, we have:

(x - 1.5)^2 = -2.75.

Then, we would take the square root of each side.  From here, we could go two ways, depending upon where you are in class.  We could say we can't take the square root of a negative number on the left, so there is no "real number" solution.  Or, if you all have had "complex numbers" yet, we can take the square root of a negative number.  We take the negative sign out and make it i:

x-1.5 = +- (i)(sqrt 2.75)                 2.75 = 11/4

x-1.5 = +- i(sqrt 11)/2

Then, we add 1.5 to each side:

x = 1.5 +- i*(sqrt 11)/2                1.5 = 3/2

x = 3/2 +- i*(sqrt 11)/2

x = (3 +- i*sqrt11)/2

The last 3 lines all the same thing, just different forms of it, not sure which form you would need.

Good luck, Rosey.  I hope this helps.

Till Then,

Steve

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