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2x^2-14*x+10=8 equation

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Numerical solution:

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The solution

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   2                
2*x  - 14*x + 10 = 8
(2x2−14x)+10=8\left(2 x^{2} - 14 x\right) + 10 = 8
Simplify
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
(2x2−14x)+10=8\left(2 x^{2} - 14 x\right) + 10 = 8
to
((2x2−14x)+10)−8=0\left(\left(2 x^{2} - 14 x\right) + 10\right) - 8 = 0
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=2a = 2
b=−14b = -14
c=2c = 2
, then
D = b^2 - 4 * a * c = 

(-14)^2 - 4 * (2) * (2) = 180

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

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

or
x1=352+72x_{1} = \frac{3 \sqrt{5}}{2} + \frac{7}{2}
x2=72−352x_{2} = \frac{7}{2} - \frac{3 \sqrt{5}}{2}
Vieta's Theorem
rewrite the equation
(2x2−14x)+10=8\left(2 x^{2} - 14 x\right) + 10 = 8
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
x2−7x+1=0x^{2} - 7 x + 1 = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−7p = -7
q=caq = \frac{c}{a}
q=1q = 1
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=7x_{1} + x_{2} = 7
x1x2=1x_{1} x_{2} = 1
The graph
05-10-510152025-500500
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
        ___           ___
7   3*\/ 5    7   3*\/ 5 
- - ------- + - + -------
2      2      2      2
(72−352)+(352+72)\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right) + \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right) 
=
7
77 
product
/        ___\ /        ___\
|7   3*\/ 5 | |7   3*\/ 5 |
|- - -------|*|- + -------|
\2      2   / \2      2   /
(72−352)(352+72)\left(\frac{7}{2} - \frac{3 \sqrt{5}}{2}\right) \left(\frac{3 \sqrt{5}}{2} + \frac{7}{2}\right) 
=
1
11 
1
Rapid solution [src] 
             ___
     7   3*\/ 5 
x1 = - - -------
     2      2
x1=72−352x_{1} = \frac{7}{2} - \frac{3 \sqrt{5}}{2} 
             ___
     7   3*\/ 5 
x2 = - + -------
     2      2
x2=352+72x_{2} = \frac{3 \sqrt{5}}{2} + \frac{7}{2} 
x2 = 3*sqrt(5)/2 + 7/2
Numerical answer [src] 
x1 = 0.145898033750315
x2 = 6.85410196624968
x2 = 6.85410196624968
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