Solve the equation x^2 - 18x + 46 = 3x

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The equation x^2 - 18x + 46 = 3x has to be solved for x.

x^2 - 18x + 46 = 3x

=> x^2 - 18x + 46 - 3x = 0

=> x^2 - 21x + 46 = 0

x1 = `(21 + sqrt(21^2 - 4*46))/2 = (21 + sqrt 257)/2`

x2 = `(21 - sqrt 257)/2`

**The solution of the equation x^2 - 18x + 46 = 3x is `(21 + sqrt 257)/2` and **`(21 - sqrt 257)/2`

Given quadratic equation is `x^2-18x+46=3x`

or, `x^2-21x+46=0` .

Let `x=a` and `x=b` be the two roots of the given equation. So we can write

`x^2-21x+46=(x-a)(x-b)`

or, `x^2-21x+46=x^2-(a+b)x+ab`

Comparing various powers of `x` we get

`(a+b)=21` (1) and `ab=46` (2).

We know that `(a-b)^2=(a+b)^2-4ab`

So, `(a-b)^2=21^2-4.46=257`

or, `(a-b)=+-sqrt257` ..............(3)

Using equation (1) and (3) we get (by considering positive sign for (a-b))

`2a=21+sqrt257`

or, `a=(21+sqrt257)/2`

and `b=(21-sqrt257)/2` .

If we take -sign for the value of (a-b) we get

`a=(21-sqrt257)/2`

and `b=(21+sqrt257)/2` .

`x^2-18x +46=3x`

subrtacting `3x` both sides:

`x^2-18x+46-3x=3x-3x`

`x^2-21x+46=0`

re-wrting:

`x^2- 2x(21/2) +441/4 - 257/4=0`

`(x-21/2)^2=257/4`

extracting square root both sides:

`x-21/2= +-1/2sqrt(257)`

so: `x=(21+-sqrt(257))/2`

`x_1=18,5156` `x_2= 2,4844`

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