Mister Exam
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- Equation:
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Equation -x^3+4*x^2+4*x-16=0
- Express {x} in terms of y where:
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Identical expressions
- (two , five *x- fifteen)*y-x= four
- (2,5 multiply by x minus 15) multiply by y minus x equally 4
- (two , five multiply by x minus fifteen) multiply by y minus x equally four
- (2,5x-15)y-x=4
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Similar expressions
- (2,5*x+15)*y-x=4
- (2,5*x-15)*y+x=4
(2,5*x-15)*y-x=4 equation
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The solution
You have entered
[src]
/5*x \ |--- - 15|*y - x = 4 \ 2 /
$$- x + y \left(\frac{5 x}{2} - 15\right) = 4$$
Detail solution
Given the equation:
Expand expressions:
Reducing, you get:
Move free summands (without x)
from left part to right part, we given:
$$\frac{5 x y}{2} - x - 15 y = 4$$
Move the summands with the other variables
from left part to right part, we given:
$$\frac{5 x y}{2} - x = 15 y + 4$$
Divide both parts of the equation by (-x + 5*x*y/2)/x
We get the answer: x = 2*(4 + 15*y)/(-2 + 5*y)
((5/2)*x-15)*y-x = 4
Expand expressions:
- 15*y + 5*x*y/2 - x = 4
Reducing, you get:
-4 - x - 15*y + 5*x*y/2 = 0
Move free summands (without x)
from left part to right part, we given:
$$\frac{5 x y}{2} - x - 15 y = 4$$
Move the summands with the other variables
from left part to right part, we given:
$$\frac{5 x y}{2} - x = 15 y + 4$$
Divide both parts of the equation by (-x + 5*x*y/2)/x
x = 4 + 15*y / ((-x + 5*x*y/2)/x)
We get the answer: x = 2*(4 + 15*y)/(-2 + 5*y)
The graph
Sum and product of roots
[src]
sum
2 / 10*(4 + 15*re(y))*im(y) 30*(-2 + 5*re(y))*im(y) \ 150*im (y) 2*(-2 + 5*re(y))*(4 + 15*re(y)) I*|- --------------------------- + ---------------------------| + --------------------------- + ------------------------------- | 2 2 2 2 | 2 2 2 2 \ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)/ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)
$$i \left(\frac{30 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{10 \left(15 \operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{2 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \left(15 \operatorname{re}{\left(y\right)} + 4\right)}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{150 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
=
2 / 10*(4 + 15*re(y))*im(y) 30*(-2 + 5*re(y))*im(y) \ 150*im (y) 2*(-2 + 5*re(y))*(4 + 15*re(y)) I*|- --------------------------- + ---------------------------| + --------------------------- + ------------------------------- | 2 2 2 2 | 2 2 2 2 \ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)/ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)
$$i \left(\frac{30 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{10 \left(15 \operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{2 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \left(15 \operatorname{re}{\left(y\right)} + 4\right)}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{150 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
product
2 / 10*(4 + 15*re(y))*im(y) 30*(-2 + 5*re(y))*im(y) \ 150*im (y) 2*(-2 + 5*re(y))*(4 + 15*re(y)) I*|- --------------------------- + ---------------------------| + --------------------------- + ------------------------------- | 2 2 2 2 | 2 2 2 2 \ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)/ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)
$$i \left(\frac{30 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{10 \left(15 \operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{2 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \left(15 \operatorname{re}{\left(y\right)} + 4\right)}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{150 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
=
/ 2 \
2*\75*im (y) + (-2 + 5*re(y))*(4 + 15*re(y)) - 50*I*im(y)/
----------------------------------------------------------
2 2
(-2 + 5*re(y)) + 25*im (y)
$$\frac{2 \left(\left(5 \operatorname{re}{\left(y\right)} - 2\right) \left(15 \operatorname{re}{\left(y\right)} + 4\right) + 75 \left(\operatorname{im}{\left(y\right)}\right)^{2} - 50 i \operatorname{im}{\left(y\right)}\right)}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
2*(75*im(y)^2 + (-2 + 5*re(y))*(4 + 15*re(y)) - 50*i*im(y))/((-2 + 5*re(y))^2 + 25*im(y)^2)
Rapid solution
[src]
2
/ 10*(4 + 15*re(y))*im(y) 30*(-2 + 5*re(y))*im(y) \ 150*im (y) 2*(-2 + 5*re(y))*(4 + 15*re(y))
x1 = I*|- --------------------------- + ---------------------------| + --------------------------- + -------------------------------
| 2 2 2 2 | 2 2 2 2
\ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)/ (-2 + 5*re(y)) + 25*im (y) (-2 + 5*re(y)) + 25*im (y)
$$x_{1} = i \left(\frac{30 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} - \frac{10 \left(15 \operatorname{re}{\left(y\right)} + 4\right) \operatorname{im}{\left(y\right)}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}\right) + \frac{2 \left(5 \operatorname{re}{\left(y\right)} - 2\right) \left(15 \operatorname{re}{\left(y\right)} + 4\right)}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}} + \frac{150 \left(\operatorname{im}{\left(y\right)}\right)^{2}}{\left(5 \operatorname{re}{\left(y\right)} - 2\right)^{2} + 25 \left(\operatorname{im}{\left(y\right)}\right)^{2}}$$
x1 = i*(30*(5*re(y) - 2)*im(y)/((5*re(y) - 2)^2 + 25*im(y)^2) - 10*(15*re(y) + 4)*im(y)/((5*re(y) - 2)^2 + 25*im(y)^2)) + 2*(5*re(y) - 2)*(15*re(y) + 4)/((5*re(y) - 2)^2 + 25*im(y)^2) + 150*im(y)^2/((5*re(y) - 2)^2 + 25*im(y)^2)