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7x^2-6x+5=0

7x^2-6x+5=0 equation

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Numerical solution:

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The solution

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   2              
7*x  - 6*x + 5 = 0
7x2−6x+5=07 x^{2} - 6 x + 5 = 0
Simplify
Detail solution
This equation is of the form
a∗x2+b∗x+c=0a*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=b2−4acD = b^2 - 4 a c is the discriminant.
Because
a=7a = 7
b=−6b = -6
c=5c = 5
, then
D=b2−4∗a∗c=D = b^2 - 4 * a * c =
(−1)7⋅4⋅5+(−6)2=−104\left(-1\right) 7 \cdot 4 \cdot 5 + \left(-6\right)^{2} = -104
Because D<0, then the equation
has no real roots,
but complex roots is exists.
x1=(−b+D)2ax_1 = \frac{(-b + \sqrt{D})}{2 a}
x2=(−b−D)2ax_2 = \frac{(-b - \sqrt{D})}{2 a}
or
x1=37+26i7x_{1} = \frac{3}{7} + \frac{\sqrt{26} i}{7}
Simplify
x2=37−26i7x_{2} = \frac{3}{7} - \frac{\sqrt{26} i}{7}
Simplify
Vieta's Theorem
rewrite the equation
7x2−6x+5=07 x^{2} - 6 x + 5 = 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
x2−6x7+57=0x^{2} - \frac{6 x}{7} + \frac{5}{7} = 0
px+x2+q=0p x + x^{2} + q = 0
where
p=bap = \frac{b}{a}
p=−67p = - \frac{6}{7}
q=caq = \frac{c}{a}
q=57q = \frac{5}{7}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=67x_{1} + x_{2} = \frac{6}{7}
x1x2=57x_{1} x_{2} = \frac{5}{7}
The graph
-1.0-0.50.00.51.01.52.02.5020
Plot the graph f1(x)
Plot the graph f2(x)
Rapid solution [src] 
              ____
      3   I*\/ 26 
x_1 = - - --------
      7      7
x1=37−26i7x_{1} = \frac{3}{7} - \frac{\sqrt{26} i}{7}
Simplify 
              ____
      3   I*\/ 26 
x_2 = - + --------
      7      7
x2=37+26i7x_{2} = \frac{3}{7} + \frac{\sqrt{26} i}{7}
Simplify
Sum and product of roots [src] 
sum
        ____           ____
3   I*\/ 26    3   I*\/ 26 
- - -------- + - + --------
7      7       7      7
(37−26i7)+(37+26i7)\left(\frac{3}{7} - \frac{\sqrt{26} i}{7}\right) + \left(\frac{3}{7} + \frac{\sqrt{26} i}{7}\right) 
=
6/7
67\frac{6}{7}
Simplify 
product
        ____           ____
3   I*\/ 26    3   I*\/ 26 
- - -------- * - + --------
7      7       7      7
(37−26i7)∗(37+26i7)\left(\frac{3}{7} - \frac{\sqrt{26} i}{7}\right) * \left(\frac{3}{7} + \frac{\sqrt{26} i}{7}\right) 
=
5/7
57\frac{5}{7}
Simplify
Numerical answer [src] 
x1 = 0.428571428571429 - 0.728431359084684*i
x2 = 0.428571428571429 + 0.728431359084684*i
x2 = 0.428571428571429 + 0.728431359084684*i
The graph
7x^2-6x+5=0 equation
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