Mister Exam
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x^2+x+a=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
$$a = 1$$
$$b = 1$$
$$c = a$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{1 - 4 a}}{2} - \frac{1}{2}$$
$$x_{2} = - \frac{\sqrt{1 - 4 a}}{2} - \frac{1}{2}$$
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
$$a = 1$$
$$b = 1$$
$$c = a$$
, then
D = b^2 - 4 * a * c =
(1)^2 - 4 * (1) * (a) = 1 - 4*a
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{1 - 4 a}}{2} - \frac{1}{2}$$
$$x_{2} = - \frac{\sqrt{1 - 4 a}}{2} - \frac{1}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = a$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = a$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 1$$
$$q = \frac{c}{a}$$
$$q = a$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = -1$$
$$x_{1} x_{2} = a$$
The graph
Rapid solution
[src]
____________________________ ____________________________
4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
\/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|
1 \ 2 / \ 2 /
x1 = - - - ----------------------------------------------------------------- - -------------------------------------------------------------------
2 2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}$$
____________________________ ____________________________
4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
\/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|
1 \ 2 / \ 2 /
x2 = - - + ----------------------------------------------------------------- + -------------------------------------------------------------------
2 2 2
$$x_{2} = \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}$$
x2 = i*((1 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*sin(atan2(-4*im(a, 1 - 4*re(a))/2)/2 + ((1 - 4*re(a))^2 + 16*im(a)^2)^(1/4)*cos(atan2(-4*im(a), 1 - 4*re(a))/2)/2 - 1/2)
Sum and product of roots
[src]
sum
____________________________ ____________________________ ____________________________ ____________________________
4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\
\/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------| \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|
1 \ 2 / \ 2 / 1 \ 2 / \ 2 /
- - - ----------------------------------------------------------------- - ------------------------------------------------------------------- + - - + ----------------------------------------------------------------- + -------------------------------------------------------------------
2 2 2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}\right) + \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}\right)$$
=
-1
$$-1$$
product
/ ____________________________ ____________________________ \ / ____________________________ ____________________________ \ | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| | 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\ 4 / 2 2 /atan2(-4*im(a), 1 - 4*re(a))\| | \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| | \/ (1 - 4*re(a)) + 16*im (a) *cos|----------------------------| I*\/ (1 - 4*re(a)) + 16*im (a) *sin|----------------------------|| | 1 \ 2 / \ 2 /| | 1 \ 2 / \ 2 /| |- - - ----------------------------------------------------------------- - -------------------------------------------------------------------|*|- - + ----------------------------------------------------------------- + -------------------------------------------------------------------| \ 2 2 2 / \ 2 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}\right) \left(\frac{i \sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(1 - 4 \operatorname{re}{\left(a\right)}\right)^{2} + 16 \left(\operatorname{im}{\left(a\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(- 4 \operatorname{im}{\left(a\right)},1 - 4 \operatorname{re}{\left(a\right)} \right)}}{2} \right)}}{2} - \frac{1}{2}\right)$$
=
I*im(a) + re(a)
$$\operatorname{re}{\left(a\right)} + i \operatorname{im}{\left(a\right)}$$
i*im(a) + re(a)