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2z=(ax+y)^2+b equation

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Numerical solution:

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The solution

Detail solution
Given the linear equation:
2*z = (a*x+y)^2+b

Expand brackets in the right part
2*z = a*x+y^2+b

Divide both parts of the equation by 2
z = b + (y + a*x)^2 / (2)

We get the answer: z = b/2 + (y + a*x)^2/2
The graph
Rapid solution [src]
                   2
      b   (y + a*x) 
z_1 = - + ----------
      2       2     
$$z_{1} = \frac{\left(a x + y\right)^{2}}{2} + \frac{b}{2}$$
Sum and product of roots [src]
sum
             2
b   (y + a*x) 
- + ----------
2       2     
$$\left(\frac{\left(a x + y\right)^{2}}{2} + \frac{b}{2}\right)$$
=
             2
b   (y + a*x) 
- + ----------
2       2     
$$\frac{\left(a x + y\right)^{2}}{2} + \frac{b}{2}$$
product
             2
b   (y + a*x) 
- + ----------
2       2     
$$\left(\frac{\left(a x + y\right)^{2}}{2} + \frac{b}{2}\right)$$
=
             2
b   (y + a*x) 
- + ----------
2       2     
$$\frac{\left(a x + y\right)^{2}}{2} + \frac{b}{2}$$