(a+2)2x-20=5a+4x equation

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(a + 2)*2*x - 20 = 5*a + 4*x

$$x 2 \left(a + 2\right) - 20 = 5 a + 4 x$$

Simplify

Detail solution

Given the linear equation:

(a+2)*2*x-20 = 5*a+4*x

Expand brackets in the left part

a*2+2*2*x-20 = 5*a+4*x

Looking for similar summands in the left part:

-20 + x*(4 + 2*a) = 5*a+4*x

Looking for similar summands in the right part:

-20 + x*(4 + 2*a) = 4*x + 5*a

Move free summands (without x)
from left part to right part, we given:
$$x \left(2 a + 4\right) = 5 a + 4 x + 20$$
Move the summands with the unknown x
from the right part to the left part:
$$x \left(2 a + 4\right) + \left(-4\right) x = 5 a + 20$$
Divide both parts of the equation by (-4*x + x*(4 + 2*a))/x

x = 20 + 5*a / ((-4*x + x*(4 + 2*a))/x)

We get the answer: x = 5/2 + 10/a

The solution of the parametric equation

Given the equation with a parameter:
$$x \left(2 a + 4\right) - 20 = 5 a + 4 x$$
Коэффициент при x равен
$$2 a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$- 2 x - 15 = 0$$
its solution
$$x = - \frac{15}{2}$$
With
$$a = 0$$
the equation
$$-20 = 0$$
its solution
no solutions

The graph

Rapid solution [src]

     5       10*re(a)         10*I*im(a)  
x1 = - + --------------- - ---------------
     2     2        2        2        2   
         im (a) + re (a)   im (a) + re (a)

$$x_{1} = \frac{5}{2} + \frac{10 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{10 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

x1 = 5/2 + 10*re(a)/(re(a)^2 + im(a)^2) - 10*i*im(a)/(re(a)^2 + im(a)^2)

Sum and product of roots [src]

sum

5       10*re(a)         10*I*im(a)  
- + --------------- - ---------------
2     2        2        2        2   
    im (a) + re (a)   im (a) + re (a)

$$\frac{5}{2} + \frac{10 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{10 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

5       10*re(a)         10*I*im(a)  
- + --------------- - ---------------
2     2        2        2        2   
    im (a) + re (a)   im (a) + re (a)

$$\frac{5}{2} + \frac{10 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{10 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

product

5       10*re(a)         10*I*im(a)  
- + --------------- - ---------------
2     2        2        2        2   
    im (a) + re (a)   im (a) + re (a)

$$\frac{5}{2} + \frac{10 \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} - \frac{10 i \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$

  /  2        2                         \
5*\im (a) + re (a) + 4*re(a) - 4*I*im(a)/
-----------------------------------------
             /  2        2   \           
           2*\im (a) + re (a)/

$$\frac{5 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + 4 \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} - 4 i \operatorname{im}{\left(a\right)}\right)}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}$$

5*(im(a)^2 + re(a)^2 + 4*re(a) - 4*i*im(a))/(2*(im(a)^2 + re(a)^2))

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(a+2)2x-20=5a+4x equation

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The solution