Mister Exam
Other calculators
-
How to use it?
- Equation:
-
Equation 12-(4-x)^2=x(3-x)
-
Equation 3^x=-3
-
Equation x^2-7*x+5=0
-
Equation (x+1)^2=0
- Express {x} in terms of y where:
- -11*x-14*y=6
- -17*x+9*y=-18
- 15*x+6*y=16
- 16*x-1*y=17
-
Identical expressions
- (3xy- one)(one +3xy)= zero
- (3xy minus 1)(1 plus 3xy) equally 0
- (3xy minus one)(one plus 3xy) equally zero
- 3xy-11+3xy=0
- (3xy-1)(1+3xy)=O
-
Similar expressions
- (3xy+1)(1+3xy)=0
- (3xy-1)(1-3xy)=0
(3xy-1)(1+3xy)=0 equation
The teacher will be very surprised to see your correct solution 😉
The solution
You have entered
[src]
(3*x*y - 1)*(1 + 3*x*y) = 0
$$\left(3 x y - 1\right) \left(3 x y + 1\right) = 0$$
Detail solution
Expand the expression in the equation
$$\left(3 x y - 1\right) \left(3 x y + 1\right) = 0$$
We get the quadratic equation
$$9 x^{2} y^{2} - 3 x y + 3 x y - 1 = 0$$
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
$$a = 9 y^{2}$$
$$b = 0$$
$$c = -1$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{y^{2}}}{3 y^{2}}$$
$$x_{2} = - \frac{\sqrt{y^{2}}}{3 y^{2}}$$
$$\left(3 x y - 1\right) \left(3 x y + 1\right) = 0$$
We get the quadratic equation
$$9 x^{2} y^{2} - 3 x y + 3 x y - 1 = 0$$
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
$$a = 9 y^{2}$$
$$b = 0$$
$$c = -1$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (9*y^2) * (-1) = 36*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{y^{2}}}{3 y^{2}}$$
$$x_{2} = - \frac{\sqrt{y^{2}}}{3 y^{2}}$$
The graph
Rapid solution
[src]
re(y) I*im(y)
x1 = - ------------------- + -------------------
/ 2 2 \ / 2 2 \
3*\im (y) + re (y)/ 3*\im (y) + re (y)/
$$x_{1} = - \frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} + \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}$$
re(y) I*im(y)
x2 = ------------------- - -------------------
/ 2 2 \ / 2 2 \
3*\im (y) + re (y)/ 3*\im (y) + re (y)/
$$x_{2} = \frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} - \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}$$
x2 = re(y)/(3*(re(y)^2 + im(y)^2)) - i*im(y)/(3*(re(y)^2 + im(y)^2))
Sum and product of roots
[src]
sum
re(y) I*im(y) re(y) I*im(y)
- ------------------- + ------------------- + ------------------- - -------------------
/ 2 2 \ / 2 2 \ / 2 2 \ / 2 2 \
3*\im (y) + re (y)/ 3*\im (y) + re (y)/ 3*\im (y) + re (y)/ 3*\im (y) + re (y)/
$$\left(- \frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} + \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}\right) + \left(\frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} - \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}\right)$$
=
0
$$0$$
product
/ re(y) I*im(y) \ / re(y) I*im(y) \ |- ------------------- + -------------------|*|------------------- - -------------------| | / 2 2 \ / 2 2 \| | / 2 2 \ / 2 2 \| \ 3*\im (y) + re (y)/ 3*\im (y) + re (y)// \3*\im (y) + re (y)/ 3*\im (y) + re (y)//
$$\left(- \frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} + \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}\right) \left(\frac{\operatorname{re}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)} - \frac{i \operatorname{im}{\left(y\right)}}{3 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)}\right)$$
=
2
-(-I*im(y) + re(y))
---------------------
2
/ 2 2 \
9*\im (y) + re (y)/
$$- \frac{\left(\operatorname{re}{\left(y\right)} - i \operatorname{im}{\left(y\right)}\right)^{2}}{9 \left(\left(\operatorname{re}{\left(y\right)}\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}\right)^{2}}$$
-(-i*im(y) + re(y))^2/(9*(im(y)^2 + re(y)^2)^2)