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x^2-3*x-3=0

x^2-3*x-3=0 equation

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Numerical solution:

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The solution

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 2              
x  - 3*x - 3 = 0
(x2−3x)−3=0\left(x^{2} - 3 x\right) - 3 = 0
Simplify
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:
x1=D−b2ax_{1} = \frac{\sqrt{D} - b}{2 a}
x2=−D−b2ax_{2} = \frac{- \sqrt{D} - b}{2 a}
where D = b^2 - 4*a*c - it is the discriminant.
Because
a=1a = 1
b=−3b = -3
c=−3c = -3
, then
D = b^2 - 4 * a * c = 

(-3)^2 - 4 * (1) * (-3) = 21

Because D > 0, then the equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)

x2 = (-b - sqrt(D)) / (2*a)

or
x1=32+212x_{1} = \frac{3}{2} + \frac{\sqrt{21}}{2}
x2=32−212x_{2} = \frac{3}{2} - \frac{\sqrt{21}}{2}
Vieta's Theorem
it is reduced quadratic equation
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−3p = -3
q=caq = \frac{c}{a}
q=−3q = -3
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=3x_{1} + x_{2} = 3
x1x2=−3x_{1} x_{2} = -3
The graph
05-15-10-5101520-200200
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
           ____
     3   \/ 21 
x1 = - - ------
     2     2
x1=32−212x_{1} = \frac{3}{2} - \frac{\sqrt{21}}{2} 
           ____
     3   \/ 21 
x2 = - + ------
     2     2
x2=32+212x_{2} = \frac{3}{2} + \frac{\sqrt{21}}{2} 
x2 = 3/2 + sqrt(21)/2
Sum and product of roots [src] 
sum
      ____         ____
3   \/ 21    3   \/ 21 
- - ------ + - + ------
2     2      2     2
(32−212)+(32+212)\left(\frac{3}{2} - \frac{\sqrt{21}}{2}\right) + \left(\frac{3}{2} + \frac{\sqrt{21}}{2}\right) 
=
3
33 
product
/      ____\ /      ____\
|3   \/ 21 | |3   \/ 21 |
|- - ------|*|- + ------|
\2     2   / \2     2   /
(32−212)(32+212)\left(\frac{3}{2} - \frac{\sqrt{21}}{2}\right) \left(\frac{3}{2} + \frac{\sqrt{21}}{2}\right) 
=
-3
−3-3 
-3
Numerical answer [src] 
x1 = 3.79128784747792
x2 = -0.79128784747792
x2 = -0.79128784747792
The graph
x^2-3*x-3=0 equation
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