Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation cos(x)=-3
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Equation x²-6x+5=0
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Equation (5+7*a)*15=-30
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Equation 1/2x=5
- Express {x} in terms of y where:
- -19*x-10*y=-10
- -3*x-18*y=-15
- 20*x-10*y=6
- 17*x-8*y=8
- Derivative of:
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2*y
- Integral of d{x}:
-
2*y
- 2*y
-
Identical expressions
- x.diff(x)+(three /y)*x= two *y
- x.diff(x) plus (3 divide by y) multiply by x equally 2 multiply by y
- x.diff(x) plus (three divide by y) multiply by x equally two multiply by y
- x.diff(x)+(3/y)x=2y
- x.diffx+3/yx=2y
- x.diff(x)+(3 divide by y)*x=2*y
-
Similar expressions
- 3(y-5)-2(y-4)=8
- sin(3x+2y)-0,3=x^2+y^2-1
- 5y+2(3-4y)=2y+21
- x.diff(x)-(3/y)*x=2*y
- 17+3y-2y=-13
- x^4-4*(x^3)*y+2*(x^2)*(y^2)-4*x*(y^3)+y^4-0,8y=0
x.diff(x)+(3/y)*x=2*y equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
//x for 0 = 1\ || | 3 |<1 for 1 = 1| + -*x = 2*y || | y \\0 otherwise/
$$x \frac{3}{y} + \begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases} = 2 y$$
Detail solution
Given the equation:
$$x \frac{3}{y} + \begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases} = 2 y$$
Multiply the equation sides by the denominators:
and y
we get:
$$y \left(x \frac{3}{y} + \begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases}\right) = y 2 y$$
$$3 x + y = 2 y^{2}$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$3 x + y = 2 y^{2}$$
to
$$3 x - 2 y^{2} + y = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 1$$
$$c = 3 x$$
, then
The equation has two roots.
or
$$y_{1} = \frac{1}{4} - \frac{\sqrt{24 x + 1}}{4}$$
$$y_{2} = \frac{\sqrt{24 x + 1}}{4} + \frac{1}{4}$$
$$x \frac{3}{y} + \begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases} = 2 y$$
Multiply the equation sides by the denominators:
and y
we get:
$$y \left(x \frac{3}{y} + \begin{cases} x & \text{for}\: 0 = 1 \\1 & \text{for}\: 1 = 1 \\0 & \text{otherwise} \end{cases}\right) = y 2 y$$
$$3 x + y = 2 y^{2}$$
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$3 x + y = 2 y^{2}$$
to
$$3 x - 2 y^{2} + y = 0$$
This equation is of the form
a*y^2 + b*y + c = 0
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$y_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$y_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = -2$$
$$b = 1$$
$$c = 3 x$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (-2) * (3*x) = 1 + 24*x
The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = \frac{1}{4} - \frac{\sqrt{24 x + 1}}{4}$$
$$y_{2} = \frac{\sqrt{24 x + 1}}{4} + \frac{1}{4}$$
The graph
Sum and product of roots
[src]
sum
______________________________ ______________________________ ______________________________ ______________________________
4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\
\/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------| \/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------|
1 \ 2 / \ 2 / 1 \ 2 / \ 2 /
- - -------------------------------------------------------------------- - ---------------------------------------------------------------------- + - + -------------------------------------------------------------------- + ----------------------------------------------------------------------
4 4 4 4 4 4
$$\left(- \frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}\right) + \left(\frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}\right)$$
=
1/2
$$\frac{1}{2}$$
product
/ ______________________________ ______________________________ \ / ______________________________ ______________________________ \ | 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\| | 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\| | \/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------|| | \/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------|| |1 \ 2 / \ 2 /| |1 \ 2 / \ 2 /| |- - -------------------------------------------------------------------- - ----------------------------------------------------------------------|*|- + -------------------------------------------------------------------- + ----------------------------------------------------------------------| \4 4 4 / \4 4 4 /
$$\left(- \frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}\right) \left(\frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}\right)$$
=
3*re(x) 3*I*im(x)
- ------- - ---------
2 2
$$- \frac{3 \operatorname{re}{\left(x\right)}}{2} - \frac{3 i \operatorname{im}{\left(x\right)}}{2}$$
-3*re(x)/2 - 3*i*im(x)/2
Rapid solution
[src]
______________________________ ______________________________
4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\
\/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------|
1 \ 2 / \ 2 /
y1 = - - -------------------------------------------------------------------- - ----------------------------------------------------------------------
4 4 4
$$y_{1} = - \frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} - \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}$$
______________________________ ______________________________
4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\ 4 / 2 2 /atan2(24*im(x), 1 + 24*re(x))\
\/ (1 + 24*re(x)) + 576*im (x) *cos|-----------------------------| I*\/ (1 + 24*re(x)) + 576*im (x) *sin|-----------------------------|
1 \ 2 / \ 2 /
y2 = - + -------------------------------------------------------------------- + ----------------------------------------------------------------------
4 4 4
$$y_{2} = \frac{i \sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{\sqrt[4]{\left(24 \operatorname{re}{\left(x\right)} + 1\right)^{2} + 576 \left(\operatorname{im}{\left(x\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(24 \operatorname{im}{\left(x\right)},24 \operatorname{re}{\left(x\right)} + 1 \right)}}{2} \right)}}{4} + \frac{1}{4}$$
y2 = i*((24*re(x) + 1)^2 + 576*im(x)^2)^(1/4)*sin(atan2(24*im(x, 24*re(x) + 1)/2)/4 + ((24*re(x) + 1)^2 + 576*im(x)^2)^(1/4)*cos(atan2(24*im(x), 24*re(x) + 1)/2)/4 + 1/4)