Mister Exam
Other calculators
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How to use it?
- Express {x} in terms of y where:
- -10*x+17*y=2
- -3*x-10*y=-2
- 6*x+13*y=-17
- -10*x-10*y=19
- Equation:
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Equation -2*(-y-2)-y-2=0
-
Equation x-1/5-x=2/9
-
Equation 1/(x-4)^2-7/x-4+10=0
-
Equation a+120=4000/5
- Plot:
-
x^2+y^2
- Canonical form:
- x^2+y^2
- Integral of d{x}:
- x^2+y^2
-
Identical expressions
- x^ two +y^ two = three xy=3
- x squared plus y squared equally 3xy equally 3
- x to the power of two plus y to the power of two equally three xy equally 3
- x2+y2=3xy=3
- x²+y²=3xy=3
- x to the power of 2+y to the power of 2=3xy=3
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Similar expressions
- x^2-y^2=3xy=3
- ((y^2-3*x*y+2*y+1,5*x+0,75)*((4-x)^0,5))/((x+6)^0,5)=x+y-2a
- (3*x*y+y)/(18*x*y^2+6*y^2)=0
- (3*x+y)*(9*x^2-3*x*y+y^2)=0
- x^2-3*x*y+2*y^2=0
Express x in terms of y where x^2+y^2=3xy=3
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
$$x^{2} + y^{2} = 3 x y$$
to
$$- 3 x y + \left(x^{2} + y^{2}\right) = 0$$
This equation is of the form
A quadratic equation can be solved
using the discriminant.
The roots of the quadratic equation:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = - 3 y$$
$$c = y^{2}$$
, then
The equation has two roots.
or
$$x_{1} = \frac{3 y}{2} + \frac{\sqrt{5} \sqrt{y^{2}}}{2}$$
$$x_{2} = \frac{3 y}{2} - \frac{\sqrt{5} \sqrt{y^{2}}}{2}$$
left part with negative sign.
The equation is transformed from
$$x^{2} + y^{2} = 3 x y$$
to
$$- 3 x y + \left(x^{2} + y^{2}\right) = 0$$
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
$$a = 1$$
$$b = - 3 y$$
$$c = y^{2}$$
, then
D = b^2 - 4 * a * c =
(-3*y)^2 - 4 * (1) * (y^2) = 5*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{3 y}{2} + \frac{\sqrt{5} \sqrt{y^{2}}}{2}$$
$$x_{2} = \frac{3 y}{2} - \frac{\sqrt{5} \sqrt{y^{2}}}{2}$$
Rapid solution
[src]
/ ___\ / ___\
|3 \/ 5 | |3 \/ 5 |
x1 = |- - -----|*re(y) + I*|- - -----|*im(y)
\2 2 / \2 2 /
$$x_{1} = \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right) \operatorname{re}{\left(y\right)} + i \left(\frac{3}{2} - \frac{\sqrt{5}}{2}\right) \operatorname{im}{\left(y\right)}$$
/ ___\ / ___\
|3 \/ 5 | |3 \/ 5 |
x2 = |- + -----|*re(y) + I*|- + -----|*im(y)
\2 2 / \2 2 /
$$x_{2} = \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right) \operatorname{re}{\left(y\right)} + i \left(\frac{\sqrt{5}}{2} + \frac{3}{2}\right) \operatorname{im}{\left(y\right)}$$
x2 = (sqrt(5)/2 + 3/2)*re(y) + i*(sqrt(5)/2 + 3/2)*im(y)