Mister Exam
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xx+2xy-yy=0 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:
$$x_{1} = \frac{\sqrt{D} - b}{2 a}$$
$$x_{2} = \frac{- \sqrt{D} - b}{2 a}$$
where D = b^2 - 4*a*c - it is the discriminant.
Because
$$a = 1$$
$$b = 2 y$$
$$c = - y^{2}$$
, then
The equation has two roots.
or
$$x_{1} = - y + \sqrt{2} \sqrt{y^{2}}$$
$$x_{2} = - y - \sqrt{2} \sqrt{y^{2}}$$
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
$$a = 1$$
$$b = 2 y$$
$$c = - y^{2}$$
, then
D = b^2 - 4 * a * c =
(2*y)^2 - 4 * (1) * (-y^2) = 8*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = - y + \sqrt{2} \sqrt{y^{2}}$$
$$x_{2} = - y - \sqrt{2} \sqrt{y^{2}}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2 y$$
$$q = \frac{c}{a}$$
$$q = - y y$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - 2 y$$
$$x_{1} x_{2} = - y y$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 2 y$$
$$q = \frac{c}{a}$$
$$q = - y y$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - 2 y$$
$$x_{1} x_{2} = - y y$$
The graph
Rapid solution
[src]
/ ___\ / ___\ x1 = \-1 + \/ 2 /*re(y) + I*\-1 + \/ 2 /*im(y)
$$x_{1} = \left(-1 + \sqrt{2}\right) \operatorname{re}{\left(y\right)} + i \left(-1 + \sqrt{2}\right) \operatorname{im}{\left(y\right)}$$
/ ___\ / ___\ x2 = \-1 - \/ 2 /*re(y) + I*\-1 - \/ 2 /*im(y)
$$x_{2} = \left(- \sqrt{2} - 1\right) \operatorname{re}{\left(y\right)} + i \left(- \sqrt{2} - 1\right) \operatorname{im}{\left(y\right)}$$
x2 = (-sqrt(2) - 1)*re(y) + i*(-sqrt(2) - 1)*im(y)
Sum and product of roots
[src]
sum
/ ___\ / ___\ / ___\ / ___\ \-1 + \/ 2 /*re(y) + I*\-1 + \/ 2 /*im(y) + \-1 - \/ 2 /*re(y) + I*\-1 - \/ 2 /*im(y)
$$\left(\left(-1 + \sqrt{2}\right) \operatorname{re}{\left(y\right)} + i \left(-1 + \sqrt{2}\right) \operatorname{im}{\left(y\right)}\right) + \left(\left(- \sqrt{2} - 1\right) \operatorname{re}{\left(y\right)} + i \left(- \sqrt{2} - 1\right) \operatorname{im}{\left(y\right)}\right)$$
=
/ ___\ / ___\ / ___\ / ___\ \-1 + \/ 2 /*re(y) + \-1 - \/ 2 /*re(y) + I*\-1 + \/ 2 /*im(y) + I*\-1 - \/ 2 /*im(y)
$$\left(- \sqrt{2} - 1\right) \operatorname{re}{\left(y\right)} + \left(-1 + \sqrt{2}\right) \operatorname{re}{\left(y\right)} + i \left(- \sqrt{2} - 1\right) \operatorname{im}{\left(y\right)} + i \left(-1 + \sqrt{2}\right) \operatorname{im}{\left(y\right)}$$
product
// ___\ / ___\ \ // ___\ / ___\ \ \\-1 + \/ 2 /*re(y) + I*\-1 + \/ 2 /*im(y)/*\\-1 - \/ 2 /*re(y) + I*\-1 - \/ 2 /*im(y)/
$$\left(\left(-1 + \sqrt{2}\right) \operatorname{re}{\left(y\right)} + i \left(-1 + \sqrt{2}\right) \operatorname{im}{\left(y\right)}\right) \left(\left(- \sqrt{2} - 1\right) \operatorname{re}{\left(y\right)} + i \left(- \sqrt{2} - 1\right) \operatorname{im}{\left(y\right)}\right)$$
=
2 2 im (y) - re (y) - 2*I*im(y)*re(y)
$$- \left(\operatorname{re}{\left(y\right)}\right)^{2} - 2 i \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)} + \left(\operatorname{im}{\left(y\right)}\right)^{2}$$
im(y)^2 - re(y)^2 - 2*i*im(y)*re(y)