SOLUTION: complete the square, and find the roots of the quadratic equation. x2 - 2/3x = 0

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Question 125713: complete the square, and find the roots of the quadratic equation.
x2 - 2/3x = 0

Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

x%5E2+-+%282%2F3%29x+=+0
substitute -2%2F3 with -0.66
then you have:

Solved by pluggable solver: COMPLETING THE SQUARE solver for quadratics
Read this lesson on completing the square by prince_abubu, if you do not know how to complete the square.
Let's convert 1x%5E2%2B-0.66x%2B0=0 to standard form by dividing both sides by 1:
We have: 1x%5E2%2B-0.66x%2B0=0. What we want to do now is to change this equation to a complete square %28x%2Bsomenumber%29%5E2+%2B+othernumber. How can we find out values of somenumber and othernumber that would make it work?
Look at %28x%2Bsomenumber%29%5E2: %28x%2Bsomenumber%29%5E2+=+x%5E2%2B2%2Asomenumber%2Ax+%2B+somenumber%5E2. Since the coefficient in our equation 1x%5E2%2Bhighlight_red%28+-0.66%29+%2A+x%2B0=0 that goes in front of x is -0.66, we know that -0.66=2*somenumber, or somenumber+=+-0.66%2F2. So, we know that our equation can be rewritten as %28x%2B-0.66%2F2%29%5E2+%2B+othernumber, and we do not yet know the other number.
We are almost there. Finding the other number is simply a matter of not making too many mistakes. We need to find 'other number' such that %28x%2B-0.66%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1x%5E2%2B-0.66x%2Bhighlight_green%28+0+%29=0.


The highlighted red part must be equal to 0 (highlighted green part).

-0.66%5E2%2F4+%2B+othernumber+=+0, or othernumber+=+0--0.66%5E2%2F4+=+-0.1089.
So, the equation converts to %28x%2B-0.66%2F2%29%5E2+%2B+-0.1089+=+0, or %28x%2B-0.66%2F2%29%5E2+=+0.1089.

Our equation converted to a square %28x%2B-0.66%2F2%29%5E2, equated to a number (0.1089).

Since the right part 0.1089 is greater than zero, there are two solutions:


, or

system%28+%28x%2B-0.66%2F2%29+=+0.33%2C+%28x%2B-0.66%2F2%29+=+-0.33+%29
system%28+x%2B-0.66%2F2+=+0.33%2C+x%2B-0.66%2F2+=+-0.33+%29
system%28+x+=+0.33--0.66%2F2%2C+x+=+-0.33--0.66%2F2+%29

system%28+x+=+0.66%2C+x+=+0+%29
Answer: x=0.66, 0.