Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^3+4*x^2-x-4=0
-
Equation (x+2)^4+(x+2)^2-12=0
-
Equation 2*cos(x)+1=0
-
Equation 8x-8=20-6x
- Express {x} in terms of y where:
- -19*x+1*y=0
- -8*x-15*y=15
- 14*x-13*y=-10
- 1*x-6*y=-19
- Canonical form:
- x^2+xy+y^2
- x^2+xy+y^2
-
Identical expressions
- x^ two +xy+y^ two
- x squared plus xy plus y squared
- x to the power of two plus xy plus y to the power of two
- x2+xy+y2
- x²+xy+y²
- x to the power of 2+xy+y to the power of 2
-
Similar expressions
- x^2-xy+y^2
- x^2+xy-y^2
x^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
$$a = 1$$
$$b = x$$
$$c = x^{2}$$
, then
The equation has two roots.
or
$$y_{1} = - \frac{x}{2} + \frac{\sqrt{3} \sqrt{- x^{2}}}{2}$$
$$y_{2} = - \frac{x}{2} - \frac{\sqrt{3} \sqrt{- x^{2}}}{2}$$
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
$$a = 1$$
$$b = x$$
$$c = x^{2}$$
, then
D = b^2 - 4 * a * c =
(x)^2 - 4 * (1) * (x^2) = -3*x^2
The equation has two roots.
y1 = (-b + sqrt(D)) / (2*a)
y2 = (-b - sqrt(D)) / (2*a)
or
$$y_{1} = - \frac{x}{2} + \frac{\sqrt{3} \sqrt{- x^{2}}}{2}$$
$$y_{2} = - \frac{x}{2} - \frac{\sqrt{3} \sqrt{- x^{2}}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = x$$
$$q = \frac{c}{a}$$
$$q = x^{2}$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = - x$$
$$y_{1} y_{2} = x^{2}$$
$$p y + q + y^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = x$$
$$q = \frac{c}{a}$$
$$q = x^{2}$$
Vieta Formulas
$$y_{1} + y_{2} = - p$$
$$y_{1} y_{2} = q$$
$$y_{1} + y_{2} = - x$$
$$y_{1} y_{2} = x^{2}$$
The graph
Rapid solution
[src]
/ ___ \ ___
re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x)
y1 = - ----- + I*|- ----- + -----------| - -----------
2 \ 2 2 / 2
$$y_{1} = i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}$$
/ ___ \ ___
re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x)
y2 = - ----- + I*|- ----- - -----------| + -----------
2 \ 2 2 / 2
$$y_{2} = i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}$$
y2 = i*(-sqrt(3)*re(x)/2 - im(x)/2) - re(x)/2 + sqrt(3)*im(x)/2
Sum and product of roots
[src]
sum
/ ___ \ ___ / ___ \ ___
re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x) re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x)
- ----- + I*|- ----- + -----------| - ----------- + - ----- + I*|- ----- - -----------| + -----------
2 \ 2 2 / 2 2 \ 2 2 / 2
$$\left(i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right) + \left(i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right)$$
=
/ ___ \ / ___ \
| im(x) \/ 3 *re(x)| | im(x) \/ 3 *re(x)|
-re(x) + I*|- ----- + -----------| + I*|- ----- - -----------|
\ 2 2 / \ 2 2 /
$$i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) + i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \operatorname{re}{\left(x\right)}$$
product
/ / ___ \ ___ \ / / ___ \ ___ \ | re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x)| | re(x) | im(x) \/ 3 *re(x)| \/ 3 *im(x)| |- ----- + I*|- ----- + -----------| - -----------|*|- ----- + I*|- ----- - -----------| + -----------| \ 2 \ 2 2 / 2 / \ 2 \ 2 2 / 2 /
$$\left(i \left(- \frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} + \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right) \left(i \left(\frac{\sqrt{3} \operatorname{re}{\left(x\right)}}{2} - \frac{\operatorname{im}{\left(x\right)}}{2}\right) - \frac{\operatorname{re}{\left(x\right)}}{2} - \frac{\sqrt{3} \operatorname{im}{\left(x\right)}}{2}\right)$$
=
2 2 re (x) - im (x) + 2*I*im(x)*re(x)
$$\left(\operatorname{re}{\left(x\right)}\right)^{2} + 2 i \operatorname{re}{\left(x\right)} \operatorname{im}{\left(x\right)} - \left(\operatorname{im}{\left(x\right)}\right)^{2}$$
re(x)^2 - im(x)^2 + 2*i*im(x)*re(x)