Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2=5
-
Equation k^2+k=0
-
Equation 5x=0
-
Equation x^2-15=2x
- Express {x} in terms of y where:
- -4*x+3*y=9
- -2*x-19*y=-11
- -5*x+15*y=-15
- -1*x+7*y=-11
-
Identical expressions
- a*x-b=c
- a multiply by x minus b equally c
- ax-b=c
-
Similar expressions
- (x-a)(x-b)(x-c)(x-d)=0
- ax-b/a+b+bx+a/a-b=a^2+b^2/a^2-b^2
- a^x-b*x=0
- a*x+b=c
a*x-b=c equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the linear equation:
Divide both parts of the equation by (-b + a*x)/x
We get the answer: x = (b + c)/a
a*x-b = c
Divide both parts of the equation by (-b + a*x)/x
x = c / ((-b + a*x)/x)
We get the answer: x = (b + c)/a
The solution of the parametric equation
Given the equation with a parameter:
$$a x - b = c$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$- b - c - x = 0$$
its solution
$$x = - b - c$$
With
$$a = 0$$
the equation
$$- b - c = 0$$
its solution
$$a x - b = c$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$- b - c - x = 0$$
its solution
$$x = - b - c$$
With
$$a = 0$$
the equation
$$- b - c = 0$$
its solution
The graph
Sum and product of roots
[src]
sum
/(im(b) + im(c))*re(a) (re(b) + re(c))*im(a)\ (im(b) + im(c))*im(a) (re(b) + re(c))*re(a) I*|--------------------- - ---------------------| + --------------------- + --------------------- | 2 2 2 2 | 2 2 2 2 \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$i \left(- \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
=
/(im(b) + im(c))*re(a) (re(b) + re(c))*im(a)\ (im(b) + im(c))*im(a) (re(b) + re(c))*re(a) I*|--------------------- - ---------------------| + --------------------- + --------------------- | 2 2 2 2 | 2 2 2 2 \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$i \left(- \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
product
/(im(b) + im(c))*re(a) (re(b) + re(c))*im(a)\ (im(b) + im(c))*im(a) (re(b) + re(c))*re(a) I*|--------------------- - ---------------------| + --------------------- + --------------------- | 2 2 2 2 | 2 2 2 2 \ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$i \left(- \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
=
I*((im(b) + im(c))*re(a) - (re(b) + re(c))*im(a)) + (im(b) + im(c))*im(a) + (re(b) + re(c))*re(a)
-------------------------------------------------------------------------------------------------
2 2
im (a) + re (a)
$$\frac{i \left(- \left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{im}{\left(a\right)} + \left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}\right) + \left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{re}{\left(a\right)} + \left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
(i*((im(b) + im(c))*re(a) - (re(b) + re(c))*im(a)) + (im(b) + im(c))*im(a) + (re(b) + re(c))*re(a))/(im(a)^2 + re(a)^2)
Rapid solution
[src]
/(im(b) + im(c))*re(a) (re(b) + re(c))*im(a)\ (im(b) + im(c))*im(a) (re(b) + re(c))*re(a)
x1 = I*|--------------------- - ---------------------| + --------------------- + ---------------------
| 2 2 2 2 | 2 2 2 2
\ im (a) + re (a) im (a) + re (a) / im (a) + re (a) im (a) + re (a)
$$x_{1} = i \left(- \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}\right) + \frac{\left(\operatorname{re}{\left(b\right)} + \operatorname{re}{\left(c\right)}\right) \operatorname{re}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} + \frac{\left(\operatorname{im}{\left(b\right)} + \operatorname{im}{\left(c\right)}\right) \operatorname{im}{\left(a\right)}}{\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}}$$
x1 = i*(-(re(b) + re(c))*im(a)/(re(a)^2 + im(a)^2) + (im(b) + im(c))*re(a)/(re(a)^2 + im(a)^2)) + (re(b) + re(c))*re(a)/(re(a)^2 + im(a)^2) + (im(b) + im(c))*im(a)/(re(a)^2 + im(a)^2)