Answers
Answer:
x2-x+1 = 0 by Completing The Square .
Subtract 1 from both side of the equation :
x2-x = -1
Now the clever bit: Take the coefficient of x , which is 1 , divide by two, giving 1/2 , and finally square it giving 1/4
Add 1/4 to both sides of the equation :
On the right hand side we have :
-1 + 1/4 or, (-1/1)+(1/4)
The common denominator of the two fractions is 4 Adding (-4/4)+(1/4) gives -3/4
So adding to both sides we finally get :
x2-x+(1/4) = -3/4
Adding 1/4 has completed the left hand side into a perfect square :
x2-x+(1/4) =
(x-(1/2)) • (x-(1/2)) =
(x-(1/2))2
Things which are equal to the same thing are also equal to one another. Since
x2-x+(1/4) = -3/4 and
x2-x+(1/4) = (x-(1/2))2
then, according to the law of transitivity,
(x-(1/2))2 = -3/4
We'll refer to this Equation as Eq. #2.2.1
The Square Root Principle says that When two things are equal, their square roots are equal.
Note that the square root of
(x-(1/2))2 is
(x-(1/2))2/2 =
(x-(1/2))1 =
x-(1/2)
Now, applying the Square Root Principle to Eq. #2.2.1 we get:
x-(1/2) = √ -3/4
Add 1/2 to both sides to obtain:
x = 1/2 + √ -3/4
In Math, i is called the imaginary unit. It satisfies i2 =-1. Both i and -i are the square roots of -1
Since a square root has two values, one positive and the other negative
x2 - x + 1 = 0
has two solutions:
x = 1/2 + √ 3/4 • i
or
x = 1/2 - √ 3/4 • i
Note that √ 3/4 can be written as
√ 3 / √ 4 which is √ 3 / 2