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(x-4²)+(x+9²)=2x² equation

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Numerical solution:

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The solution

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                     2
x - 16 + x + 81 = 2*x
(x−16)+(x+81)=2x2\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}
Simplify
Detail solution
Move right part of the equation to
left part with negative sign.

The equation is transformed from
(x−16)+(x+81)=2x2\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}
to
−2x2+((x−16)+(x+81))=0- 2 x^{2} + \left(\left(x - 16\right) + \left(x + 81\right)\right) = 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=2b = 2
c=65c = 65
, then
D = b^2 - 4 * a * c = 

(2)^2 - 4 * (-2) * (65) = 524

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

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

or
x1=12−1312x_{1} = \frac{1}{2} - \frac{\sqrt{131}}{2}
x2=12+1312x_{2} = \frac{1}{2} + \frac{\sqrt{131}}{2}
Vieta's Theorem
rewrite the equation
(x−16)+(x+81)=2x2\left(x - 16\right) + \left(x + 81\right) = 2 x^{2}
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−x−652=0x^{2} - x - \frac{65}{2} = 0
px+q+x2=0p x + q + x^{2} = 0
where
p=bap = \frac{b}{a}
p=−1p = -1
q=caq = \frac{c}{a}
q=−652q = - \frac{65}{2}
Vieta Formulas
x1+x2=−px_{1} + x_{2} = - p
x1x2=qx_{1} x_{2} = q
x1+x2=1x_{1} + x_{2} = 1
x1x2=−652x_{1} x_{2} = - \frac{65}{2}
The graph
-15.0-12.5-10.0-7.5-5.0-2.50.02.55.07.510.012.515.001000
Plot the graph f1(x)
Plot the graph f2(x)
Sum and product of roots [src] 
sum
      _____         _____
1   \/ 131    1   \/ 131 
- - ------- + - + -------
2      2      2      2
(12−1312)+(12+1312)\left(\frac{1}{2} - \frac{\sqrt{131}}{2}\right) + \left(\frac{1}{2} + \frac{\sqrt{131}}{2}\right) 
=
1
11 
product
/      _____\ /      _____\
|1   \/ 131 | |1   \/ 131 |
|- - -------|*|- + -------|
\2      2   / \2      2   /
(12−1312)(12+1312)\left(\frac{1}{2} - \frac{\sqrt{131}}{2}\right) \left(\frac{1}{2} + \frac{\sqrt{131}}{2}\right) 
=
-65/2
−652- \frac{65}{2} 
-65/2
Rapid solution [src] 
           _____
     1   \/ 131 
x1 = - - -------
     2      2
x1=12−1312x_{1} = \frac{1}{2} - \frac{\sqrt{131}}{2} 
           _____
     1   \/ 131 
x2 = - + -------
     2      2
x2=12+1312x_{2} = \frac{1}{2} + \frac{\sqrt{131}}{2} 
x2 = 1/2 + sqrt(131)/2
Numerical answer [src] 
x1 = -5.2227615711298
x2 = 6.2227615711298
x2 = 6.2227615711298
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