Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation cos(x)=√3/2
- Equation x^2-x-56=0
- Equation f*(x)=3
-
Equation x^3=-8
- Express {x} in terms of y where:
- 3*x+10*y=-13
- 14*x+5*y=-18
- -4*x+18*y=-6
- 6*x-14*y=-9
- Derivative of:
- f*(x)
-
Identical expressions
- f*(x)= three
- f multiply by (x) equally 3
- f multiply by (x) equally three
- f(x)=3
- fx=3
f*(x)=3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Given the linear equation:
Expand brackets in the left part
Divide both parts of the equation by f
We get the answer: x = 3/f
f*(x) = 3
Expand brackets in the left part
fx = 3
Divide both parts of the equation by f
x = 3 / (f)
We get the answer: x = 3/f
The solution of the parametric equation
Given the equation with a parameter:
$$f x = 3$$
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
$$- x - 3 = 0$$
its solution
$$x = -3$$
With
$$f = 0$$
the equation
$$-3 = 0$$
its solution
no solutions
$$f x = 3$$
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
$$- x - 3 = 0$$
its solution
$$x = -3$$
With
$$f = 0$$
the equation
$$-3 = 0$$
its solution
no solutions
The graph
Rapid solution
[src]
3*re(f) 3*I*im(f)
x1 = --------------- - ---------------
2 2 2 2
im (f) + re (f) im (f) + re (f)
$$x_{1} = \frac{3 \operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{3 i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
x1 = 3*re(f)/(re(f)^2 + im(f)^2) - 3*i*im(f)/(re(f)^2 + im(f)^2)
Sum and product of roots
[src]
sum
3*re(f) 3*I*im(f) --------------- - --------------- 2 2 2 2 im (f) + re (f) im (f) + re (f)
$$\frac{3 \operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{3 i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
=
3*re(f) 3*I*im(f) --------------- - --------------- 2 2 2 2 im (f) + re (f) im (f) + re (f)
$$\frac{3 \operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{3 i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
product
3*re(f) 3*I*im(f) --------------- - --------------- 2 2 2 2 im (f) + re (f) im (f) + re (f)
$$\frac{3 \operatorname{re}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}} - \frac{3 i \operatorname{im}{\left(f\right)}}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
=
3*(-I*im(f) + re(f))
--------------------
2 2
im (f) + re (f)
$$\frac{3 \left(\operatorname{re}{\left(f\right)} - i \operatorname{im}{\left(f\right)}\right)}{\left(\operatorname{re}{\left(f\right)}\right)^{2} + \left(\operatorname{im}{\left(f\right)}\right)^{2}}$$
3*(-i*im(f) + re(f))/(im(f)^2 + re(f)^2)