Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x*7=56
-
Equation x^2=-5
-
Equation x*x=x
-
Equation 9x-4=10x
- Express {x} in terms of y where:
- -16*x-9*y=3
- 12*x+17*y=-13
- -9*x-6*y=-4
- -11*x+11*y=2
-
Identical expressions
- f*(x)*x= one
- f multiply by (x) multiply by x equally 1
- f multiply by (x) multiply by x equally one
- f(x)x=1
- fxx=1
-
Similar expressions
- f.diff(x)*(x)=2*x^2-4*x-9
f*(x)*x=1 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$x f x = 1$$
to
$$x f x - 1 = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = f$$
$$b = 0$$
$$c = -1$$
, then
The equation has two roots.
or
$$x_{1} = \frac{1}{\sqrt{f}}$$
$$x_{2} = - \frac{1}{\sqrt{f}}$$
left part with negative sign.
The equation is transformed from
$$x f x = 1$$
to
$$x f x - 1 = 0$$
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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = f$$
$$b = 0$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (f) * (-1) = 4*f
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{1}{\sqrt{f}}$$
$$x_{2} = - \frac{1}{\sqrt{f}}$$
The solution of the parametric equation
Given the equation with a parameter:
$$f x^{2} = 1$$
Коэффициент при x равен
$$f$$
then possible cases for f :
$$f < 0$$
$$f = 0$$
Consider all cases in more detail:
With
$$f < 0$$
the equation
$$- x^{2} - 1 = 0$$
its solution
no solutions
With
$$f = 0$$
the equation
$$-1 = 0$$
its solution
no solutions
$$f x^{2} = 1$$
Коэффициент при x равен
$$f$$
then possible cases for f :
$$f < 0$$
$$f = 0$$
Consider all cases in more detail:
With
$$f < 0$$
the equation
$$- x^{2} - 1 = 0$$
its solution
no solutions
With
$$f = 0$$
the equation
$$-1 = 0$$
its solution
no solutions
Vieta's Theorem
rewrite the equation
$$x f x = 1$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{f x^{2} - 1}{f} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{f}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{1}{f}$$
$$x f x = 1$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{f x^{2} - 1}{f} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{1}{f}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{1}{f}$$
The graph
Sum and product of roots
[src]
sum
/ / -im(f) re(f) \\ / / -im(f) re(f) \\ / / -im(f) re(f) \\ / / -im(f) re(f) \\
_________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------||
/ 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 ||
/ im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/|
- / ------------------ + ------------------ *cos|---------------------------------------| - I* / ------------------ + ------------------ *sin|---------------------------------------| + / ------------------ + ------------------ *cos|---------------------------------------| + I* / ------------------ + ------------------ *sin|---------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 / / 2 2 \ 2 / / 2 2 \ 2 /
4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \
\/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/
$$\left(- i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}\right)$$
=
0
$$0$$
product
/ / / -im(f) re(f) \\ / / -im(f) re(f) \\\ / / / -im(f) re(f) \\ / / -im(f) re(f) \\\ | _________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------||| | _________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------||| | / 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 ||| | / 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 ||| | / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/|| | / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/|| |- / ------------------ + ------------------ *cos|---------------------------------------| - I* / ------------------ + ------------------ *sin|---------------------------------------||*| / ------------------ + ------------------ *cos|---------------------------------------| + I* / ------------------ + ------------------ *sin|---------------------------------------|| | / 2 2 \ 2 / / 2 2 \ 2 /| | / 2 2 \ 2 / / 2 2 \ 2 /| | 4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \ | |4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \ | \ \/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/ / \\/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/ /
$$\left(- i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}\right)$$
=
/ -im(f) re(f) \
I*atan2|---------------, ---------------|
| 2 2 2 2 |
\im (f) + re (f) im (f) + re (f)/
-e
--------------------------------------------
_________________
/ 2 2
\/ im (f) + re (f)
$$- \frac{e^{i \operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}}{\sqrt{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}}$$
-exp(i*atan2(-im(f)/(im(f)^2 + re(f)^2), re(f)/(im(f)^2 + re(f)^2)))/sqrt(im(f)^2 + re(f)^2)
Rapid solution
[src]
/ / -im(f) re(f) \\ / / -im(f) re(f) \\
_________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------||
/ 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 ||
/ im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/|
x1 = - / ------------------ + ------------------ *cos|---------------------------------------| - I* / ------------------ + ------------------ *sin|---------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 /
4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \
\/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/
$$x_{1} = - i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} - \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}$$
/ / -im(f) re(f) \\ / / -im(f) re(f) \\
_________________________________________ |atan2|---------------, ---------------|| _________________________________________ |atan2|---------------, ---------------||
/ 2 2 | | 2 2 2 2 || / 2 2 | | 2 2 2 2 ||
/ im (f) re (f) | \im (f) + re (f) im (f) + re (f)/| / im (f) re (f) | \im (f) + re (f) im (f) + re (f)/|
x2 = / ------------------ + ------------------ *cos|---------------------------------------| + I* / ------------------ + ------------------ *sin|---------------------------------------|
/ 2 2 \ 2 / / 2 2 \ 2 /
4 / / 2 2 \ / 2 2 \ 4 / / 2 2 \ / 2 2 \
\/ \im (f) + re (f)/ \im (f) + re (f)/ \/ \im (f) + re (f)/ \im (f) + re (f)/
$$x_{2} = i \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)} + \sqrt[4]{\frac{\left(\operatorname{re}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}} + \frac{\left(\operatorname{im}{\left(f\right)}\right)^{2}}{\left(\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}\right)^{2}}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- \frac{\operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}},\frac{\operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} \right)}}{2} \right)}$$
x2 = i*(re(f)^2/(re(f)^2 + im(f)^2)^2 + im(f)^2/(re(f)^2 + im(f)^2)^2)^(1/4)*sin(atan2(-im(f)/(re(f)^2 + im(f)^2, re(f)/(re(f)^2 + im(f)^2))/2) + (re(f)^2/(re(f)^2 + im(f)^2)^2 + im(f)^2/(re(f)^2 + im(f)^2)^2)^(1/4)*cos(atan2(-im(f)/(re(f)^2 + im(f)^2), re(f)/(re(f)^2 + im(f)^2))/2))