Question

$x {}^{2} - x + 1$
​

Answer 1

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