Question

Part E
Solve the equation x^2- 4x + 6 = 0 by completing the square.​

Answer 1

Roots for the given polynomial p(x) is x = (2 + √2i) or x = (2 - √2i)

Explanation:

Here, the given polynomial is
`P(x) = x^(2)  - 4x + 6 = 0`

Now, in COMPLETING THE SQUARE there are various steps.

Step (1): HALF THE COEFFICIENT OF x

Here, the coefficient of x = (4) so , the half of 4 = 4/ 2 = 2

Step(2) :Square it ADD IT ON BOTH SIDES OF EQUATION

The square of 2 is
`(2)^2`.

Adding it on both sides of the polynomial, we get

`P(x) :  x^(2)  - 4x + 6  + (2)^2 = 0   + (2)^2`

Step (3): Use the ALGEBRAIC IDENTITY and make a complete square.

Now, using the identity
`(a\pm b)^2 = a^2 + b^2 \pm 2ab`

`\implies x^(2)  - 4x + 6  + (2)^2 = 0   + (2)^2\\=  (x^(2)  - 4x + (2)^2 )+ 6   = 0   + (2)^2\\= (x-2)^2   +  6  -  (2)^2 =  0\\\implies  (x-2)^2  + 2 = 0`

or,
`(x-2)^2  =  -2`

Rooting both sides, we get

`\pm (x-2)  = (\sqrt2i)`

Solving this, further,we get

x = 2 + √2i , or x = x = 2 - √2i

Hence, roots for the given polynomial p(x) is x = 2 + √2i or x = 2 - √2i