SOLUTION: Use completing the square or the quadradic formula to find the solutions to the equation: x^2-4x+2=0. Thank you for your time.

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Question 43566: Use completing the square or the quadradic formula to find the solutions to the equation: x^2-4x+2=0. Thank you for your time.
Answer by aaaaaaaa(138) About Me  (Show Source):
You can put this solution on YOUR website!
Solved by pluggable solver: SOLVE quadratic equation with variable
Quadratic equation ax%5E2%2Bbx%2Bc=0 (in our case 1x%5E2%2B-4x%2B2+=+0) has the following solutons:

x%5B12%5D+=+%28b%2B-sqrt%28+b%5E2-4ac+%29%29%2F2%5Ca

For these solutions to exist, the discriminant b%5E2-4ac should not be a negative number.

First, we need to compute the discriminant b%5E2-4ac: b%5E2-4ac=%28-4%29%5E2-4%2A1%2A2=8.

Discriminant d=8 is greater than zero. That means that there are two solutions: +x%5B12%5D+=+%28--4%2B-sqrt%28+8+%29%29%2F2%5Ca.

x%5B1%5D+=+%28-%28-4%29%2Bsqrt%28+8+%29%29%2F2%5C1+=+3.41421356237309
x%5B2%5D+=+%28-%28-4%29-sqrt%28+8+%29%29%2F2%5C1+=+0.585786437626905

Quadratic expression 1x%5E2%2B-4x%2B2 can be factored:
1x%5E2%2B-4x%2B2+=+1%28x-3.41421356237309%29%2A%28x-0.585786437626905%29
Again, the answer is: 3.41421356237309, 0.585786437626905. Here's your graph:
graph%28+500%2C+500%2C+-10%2C+10%2C+-20%2C+20%2C+1%2Ax%5E2%2B-4%2Ax%2B2+%29


Now for the other method:
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-4x%2B2=0 to standard form by dividing both sides by 1:
We have: 1x%5E2%2B-4x%2B2=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+-4%29+%2A+x%2B2=0 that goes in front of x is -4, we know that -4=2*somenumber, or somenumber+=+-4%2F2. So, we know that our equation can be rewritten as %28x%2B-4%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-4%2F2%29%5E2+%2B+othernumber is equivalent to our original equation 1x%5E2%2B-4x%2Bhighlight_green%28+2+%29=0.


The highlighted red part must be equal to 2 (highlighted green part).

-4%5E2%2F4+%2B+othernumber+=+2, or othernumber+=+2--4%5E2%2F4+=+-2.
So, the equation converts to %28x%2B-4%2F2%29%5E2+%2B+-2+=+0, or %28x%2B-4%2F2%29%5E2+=+2.

Our equation converted to a square %28x%2B-4%2F2%29%5E2, equated to a number (2).

Since the right part 2 is greater than zero, there are two solutions:

system%28+%28x%2B-4%2F2%29+=+%2Bsqrt%28+2+%29%2C+%28x%2B-4%2F2%29+=+-sqrt%28+2+%29+%29
, or

system%28+%28x%2B-4%2F2%29+=+1.4142135623731%2C+%28x%2B-4%2F2%29+=+-1.4142135623731+%29
system%28+x%2B-4%2F2+=+1.4142135623731%2C+x%2B-4%2F2+=+-1.4142135623731+%29
system%28+x+=+1.4142135623731--4%2F2%2C+x+=+-1.4142135623731--4%2F2+%29

system%28+x+=+3.41421356237309%2C+x+=+0.585786437626905+%29
Answer: x=3.41421356237309, 0.585786437626905.


Note that it says that x+=+2+%2B-+sqrt%282%29