Mister Exam
Other calculators
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How to use it?
- Express {x} in terms of y where:
- a^2*(2*x+3*y)=0
- a*x^5-b*x^3*y^2-c*x^2*y^3+d*y^5=0
- (x+7)/(x^2-2*x+3)=y
- sin(3x)^2*sqrt(16-x^2)=y
- Equation:
-
Equation (x+3)^4-4(x+3)^2-5=0
-
Equation 2x^2-x-3=0
-
Equation 9x-4x+39=94
-
Equation 2(8x-7)=18-4(5-4x)
- Canonical form:
-
y^2+x^2
- Integral of d{x}:
- y^2+x^2
- The double integral of:
- y^2+x^2
- 2x+4y
-
Identical expressions
- y^ two +x^ two =2x+4y
- y squared plus x squared equally 2x plus 4y
- y to the power of two plus x to the power of two equally 2x plus 4y
- y2+x2=2x+4y
- y²+x²=2x+4y
- y to the power of 2+x to the power of 2=2x+4y
-
Similar expressions
- (12*x+4*y)/(9*x^2-y^2)=0
- y^2+x^2=2x-4y
- y^2-x^2=2x+4y
Express y in terms of y where y^2+x^2=2x+4y
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
$$x^{2} + y^{2} = 2 x + 4 y$$
to
$$\left(- 2 x - 4 y\right) + \left(x^{2} + y^{2}\right) = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = x^{2} - 2 x$$
, then
The equation has two roots.
or
$$y_{1} = \frac{\sqrt{- 4 x^{2} + 8 x + 16}}{2} + 2$$
$$y_{2} = 2 - \frac{\sqrt{- 4 x^{2} + 8 x + 16}}{2}$$
left part with negative sign.
The equation is transformed from
$$x^{2} + y^{2} = 2 x + 4 y$$
to
$$\left(- 2 x - 4 y\right) + \left(x^{2} + y^{2}\right) = 0$$
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
$$a = 1$$
$$b = -4$$
$$c = x^{2} - 2 x$$
, then
D = b^2 - 4 * a * c =
(-4)^2 - 4 * (1) * (x^2 - 2*x) = 16 - 4*x^2 + 8*x
The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = \frac{\sqrt{- 4 x^{2} + 8 x + 16}}{2} + 2$$
$$y_{2} = 2 - \frac{\sqrt{- 4 x^{2} + 8 x + 16}}{2}$$
Rapid solution
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\2*im(x) - 2*im(x)*re(x), 4 + im (x) - re (x) + 2*re(x)/| 4 / 2 / 2 2 \ |atan2\2*im(x) - 2*im(x)*re(x), 4 + im (x) - re (x) + 2*re(x)/|
y1 = 2 - \/ (2*im(x) - 2*im(x)*re(x)) + \4 + im (x) - re (x) + 2*re(x)/ *cos|-------------------------------------------------------------| - I*\/ (2*im(x) - 2*im(x)*re(x)) + \4 + im (x) - re (x) + 2*re(x)/ *sin|-------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{1} = - i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)} + 2$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / 2 / 2 2 \ |atan2\2*im(x) - 2*im(x)*re(x), 4 + im (x) - re (x) + 2*re(x)/| 4 / 2 / 2 2 \ |atan2\2*im(x) - 2*im(x)*re(x), 4 + im (x) - re (x) + 2*re(x)/|
y2 = 2 + \/ (2*im(x) - 2*im(x)*re(x)) + \4 + im (x) - re (x) + 2*re(x)/ *cos|-------------------------------------------------------------| + I*\/ (2*im(x) - 2*im(x)*re(x)) + \4 + im (x) - re (x) + 2*re(x)/ *sin|-------------------------------------------------------------|
\ 2 / \ 2 /
$$y_{2} = i \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)}\right)^{2} + \left(- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 2 \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} + 2 \operatorname{im}{\left(x\right)},- \left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 \operatorname{re}{\left(x\right)} + \left(\operatorname{im}{\left(x\right)}\right)^{2} + 4 \right)}}{2} \right)} + 2$$
y2 = i*((-2*re(x)*im(x) + 2*im(x))^2 + (-re(x)^2 + 2*re(x) + im(x)^2 + 4)^2)^(1/4)*sin(atan2(-2*re(x)*im(x) + 2*im(x, -re(x)^2 + 2*re(x) + im(x)^2 + 4)/2) + ((-2*re(x)*im(x) + 2*im(x))^2 + (-re(x)^2 + 2*re(x) + im(x)^2 + 4)^2)^(1/4)*cos(atan2(-2*re(x)*im(x) + 2*im(x), -re(x)^2 + 2*re(x) + im(x)^2 + 4)/2) + 2)