x^2-6*x+3=0 equation
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The solution
Detail solution
This equation is of the form
a*x^2 + b*x + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = -6$$
$$c = 3$$
, then
D = b^2 - 4 * a * c =
(-6)^2 - 4 * (1) * (3) = 24
Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \sqrt{6} + 3$$
$$x_{2} = 3 - \sqrt{6}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -6$$
$$q = \frac{c}{a}$$
$$q = 3$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 6$$
$$x_{1} x_{2} = 3$$
Sum and product of roots
[src]
___ ___
3 - \/ 6 + 3 + \/ 6
$$\left(3 - \sqrt{6}\right) + \left(\sqrt{6} + 3\right)$$
$$6$$
/ ___\ / ___\
\3 - \/ 6 /*\3 + \/ 6 /
$$\left(3 - \sqrt{6}\right) \left(\sqrt{6} + 3\right)$$
$$3$$
$$x_{1} = 3 - \sqrt{6}$$
$$x_{2} = \sqrt{6} + 3$$