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x^2-x-7=0

x^2-x-7=0 equation

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Numerical solution:

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The solution

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 2            
x  - x - 7 = 0
$$\left(x^{2} - x\right) - 7 = 0$$
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
$$a = 1$$
$$b = -1$$
$$c = -7$$
, then
D = b^2 - 4 * a * c = 

(-1)^2 - 4 * (1) * (-7) = 29

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

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

or
$$x_{1} = \frac{1}{2} + \frac{\sqrt{29}}{2}$$
$$x_{2} = \frac{1}{2} - \frac{\sqrt{29}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = -1$$
$$q = \frac{c}{a}$$
$$q = -7$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 1$$
$$x_{1} x_{2} = -7$$
The graph
Sum and product of roots [src]
sum
      ____         ____
1   \/ 29    1   \/ 29 
- - ------ + - + ------
2     2      2     2   
$$\left(\frac{1}{2} - \frac{\sqrt{29}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{29}}{2}\right)$$
=
1
$$1$$
product
/      ____\ /      ____\
|1   \/ 29 | |1   \/ 29 |
|- - ------|*|- + ------|
\2     2   / \2     2   /
$$\left(\frac{1}{2} - \frac{\sqrt{29}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{29}}{2}\right)$$
=
-7
$$-7$$
-7
Rapid solution [src]
           ____
     1   \/ 29 
x1 = - - ------
     2     2   
$$x_{1} = \frac{1}{2} - \frac{\sqrt{29}}{2}$$
           ____
     1   \/ 29 
x2 = - + ------
     2     2   
$$x_{2} = \frac{1}{2} + \frac{\sqrt{29}}{2}$$
x2 = 1/2 + sqrt(29)/2
Numerical answer [src]
x1 = -2.19258240356725
x2 = 3.19258240356725
x2 = 3.19258240356725
The graph
x^2-x-7=0 equation