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-7x^2-13x+8=0 equation

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Numerical solution:

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The solution

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     2               
- 7*x  - 13*x + 8 = 0
(−7x2−13x)+8=0\left(- 7 x^{2} - 13 x\right) + 8 = 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=−7a = -7
b=−13b = -13
c=8c = 8
, then
D = b^2 - 4 * a * c = 

(-13)^2 - 4 * (-7) * (8) = 393

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

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

or
x1=−39314−1314x_{1} = - \frac{\sqrt{393}}{14} - \frac{13}{14}
x2=−1314+39314x_{2} = - \frac{13}{14} + \frac{\sqrt{393}}{14}
Vieta's Theorem
rewrite the equation
(−7x2−13x)+8=0\left(- 7 x^{2} - 13 x\right) + 8 = 0
of
ax2+bx+c=0a x^{2} + b x + c = 0
as reduced quadratic equation
x2+bxa+ca=0x^{2} + \frac{b x}{a} + \frac{c}{a} = 0
x2+13x7−87=0x^{2} + \frac{13 x}{7} - \frac{8}{7} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=137p = \frac{13}{7}
q=caq = \frac{c}{a}
q=−87q = - \frac{8}{7}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=−137x_{1} + x_{2} = - \frac{13}{7}
x1x2=−87x_{1} x_{2} = - \frac{8}{7}
The graph
05-15-10-51015-10001000
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
              _____
       13   \/ 393 
x1 = - -- + -------
       14      14
x1=−1314+39314x_{1} = - \frac{13}{14} + \frac{\sqrt{393}}{14} 
              _____
       13   \/ 393 
x2 = - -- - -------
       14      14
x2=−39314−1314x_{2} = - \frac{\sqrt{393}}{14} - \frac{13}{14} 
x2 = -sqrt(393)/14 - 13/14
Sum and product of roots [src] 
sum
         _____            _____
  13   \/ 393      13   \/ 393 
- -- + ------- + - -- - -------
  14      14       14      14
(−39314−1314)+(−1314+39314)\left(- \frac{\sqrt{393}}{14} - \frac{13}{14}\right) + \left(- \frac{13}{14} + \frac{\sqrt{393}}{14}\right) 
=
-13/7
−137- \frac{13}{7} 
product
/         _____\ /         _____\
|  13   \/ 393 | |  13   \/ 393 |
|- -- + -------|*|- -- - -------|
\  14      14  / \  14      14  /
(−1314+39314)(−39314−1314)\left(- \frac{13}{14} + \frac{\sqrt{393}}{14}\right) \left(- \frac{\sqrt{393}}{14} - \frac{13}{14}\right) 
=
-8/7
−87- \frac{8}{7} 
-8/7
Numerical answer [src] 
x1 = 0.487444828685643
x2 = -2.3445876858285
x2 = -2.3445876858285
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