Question
"I got on a test question that:
3x^2 - 2x + 7 = 0, then ( x - 1/3 )^2 = ???

The correct is -20/9.  I don't see how this is possible since you get a negative number under the radical sign (-80) when using the quadratic formula.
Could you explain this? Thanks...."

What you are discovering are imaginary numbers. These are legitimate numbers that crop up in many situations. They can be written as a real number times the square root of negative 1, which is designated

i = √-1

where the "i" stands for imaginary.

For your problem, the number under the square root sign in the quadratic equation can be written as √-80 = √(-1)(80) = i√80. The rest of the quadratic solution is

x = 1/3 ± i(√80)/6.

Subtracting 1/3 and taking the 6 into the square root gives

(x-1/3)^2 = [ i(80/36)^1/2 ]^2 = i^2(80/36) = (-1)(20/9).

Imaginary numbers are a little strange but they do exist and are important in mathematics. Don't worry if you have trouble understanding them, it took mathematicians a long time to accept them, too.
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#### randy patton

##### Expertise

college mathematics, applied math, advanced calculus, complex analysis, linear and abstract algebra, probability theory, signal processing, undergraduate physics, physical oceanography

##### Experience

26 years as a professional scientist conducting academic quality research on mostly classified projects involving math/physics modeling and simulation, data analysis and signal processing, instrument development; often ocean related

Publications
J. Physical Oceanography, 1984 "A Numerical Model for Low-Frequency Equatorial Dynamics", with M. Cane

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M.S. MIT Physical Oceanography, B.S. UC Berkeley Applied Math

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Also an Expert in Oceanography