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