Mister Exam
Other calculators
-
How to use it?
- Equation:
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Equation (y^4+4*y^2-12)/4=0
-
Equation -x=5
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Equation x^2+7x-18=0
- Equation x-y+2=0
- Express {x} in terms of y where:
- 13*x-18*y=-17
- -18*x+12*y=-8
- -14*x+4*y=-16
- 9*x-1*y=17
- xy-y^2
-
Identical expressions
- xy-y^ two
- xy minus y squared
- xy minus y to the power of two
- xy-y2
- xy-y²
- xy-y to the power of 2
-
Similar expressions
- xy+y^2
xy-y^2 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
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
$$a = -1$$
$$b = x$$
$$c = 0$$
, then
The equation has two roots.
or
$$y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2}}}{2}$$
$$y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2}}}{2}$$
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
$$a = -1$$
$$b = x$$
$$c = 0$$
, then
D = b^2 - 4 * a * c =
(x)^2 - 4 * (-1) * (0) = x^2
The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = \frac{x}{2} - \frac{\sqrt{x^{2}}}{2}$$
$$y_{2} = \frac{x}{2} + \frac{\sqrt{x^{2}}}{2}$$
Vieta's Theorem
rewrite the equation
$$x y - y^{2} = 0$$
of
$$a y^{2} + b y + c = 0$$
as reduced quadratic equation
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x y + y^{2} = 0$$
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - x$$
$$q = \frac{c}{a}$$
$$q = 0$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x$$
$$y_{1} y_{2} = 0$$
$$x y - y^{2} = 0$$
of
$$a y^{2} + b y + c = 0$$
as reduced quadratic equation
$$y^{2} + \frac{b y}{a} + \frac{c}{a} = 0$$
$$- x y + y^{2} = 0$$
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = - x$$
$$q = \frac{c}{a}$$
$$q = 0$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = x$$
$$y_{1} y_{2} = 0$$
The graph
Sum and product of roots
[src]
sum
I*im(x) + re(x)
$$\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)}$$
=
I*im(x) + re(x)
$$\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)}$$
product
0*(I*im(x) + re(x))
$$0 \left(\operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)}\right)$$
=
0
$$0$$
0
Rapid solution
[src]
y1 = 0
$$y_{1} = 0$$
y2 = I*im(x) + re(x)
$$y_{2} = \operatorname{re}{\left(x\right)} + i \operatorname{im}{\left(x\right)}$$
y2 = re(x) + i*im(x)