Mister Exam
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How to use it?
- Equation:
- Equation ax=b
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Equation x^2-x+8=0
- Equation x^2+y^2=1
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Equation x^2-8x+16=0
- Express {x} in terms of y where:
- -4*x+7*y=-16
- -1*x-20*y=1
- 14*x-10*y=-4
- -11*x+16*y=3
- Factor polynomial:
- a*x^2+b*x+c
-
Identical expressions
- a*x^ two +b*x+c= zero
- a multiply by x squared plus b multiply by x plus c equally 0
- a multiply by x to the power of two plus b multiply by x plus c equally zero
- a*x2+b*x+c=0
- a*x²+b*x+c=0
- a*x to the power of 2+b*x+c=0
- ax^2+bx+c=0
- ax2+bx+c=0
- a*x^2+b*x+c=O
-
Similar expressions
- a*x^2-b*x+c=0
- a*x^2+b*x-c=0
a*x^2+b*x+c=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
, then
The equation has two roots.
or
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$
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
True
True
True
, then
D = b^2 - 4 * a * c =
(b)^2 - 4 * (a) * (c) = b^2 - 4*a*c
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{- b + \sqrt{- 4 a c + b^{2}}}{2 a}$$
$$x_{2} = \frac{- b - \sqrt{- 4 a c + b^{2}}}{2 a}$$
Vieta's Theorem
rewrite the equation
$$c + \left(a x^{2} + b x\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + b x + c}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{b}{a}$$
$$x_{1} x_{2} = \frac{c}{a}$$
$$c + \left(a x^{2} + b x\right) = 0$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} + b x + c}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = \frac{b}{a}$$
$$q = \frac{c}{a}$$
$$q = \frac{c}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = - \frac{b}{a}$$
$$x_{1} x_{2} = \frac{c}{a}$$
The solution of the parametric equation
Given the equation with a parameter:
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$b x + c - x^{2} = 0$$
its solution
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
With
$$a = 0$$
the equation
$$b x + c = 0$$
its solution
$$x = - \frac{c}{b}$$
$$a x^{2} + b x + c = 0$$
Коэффициент при x равен
$$a$$
then possible cases for a :
$$a < 0$$
$$a = 0$$
Consider all cases in more detail:
With
$$a < 0$$
the equation
$$b x + c - x^{2} = 0$$
its solution
$$x = \frac{b}{2} - \frac{\sqrt{b^{2} + 4 c}}{2}$$
$$x = \frac{b}{2} + \frac{\sqrt{b^{2} + 4 c}}{2}$$
With
$$a = 0$$
the equation
$$b x + c = 0$$
its solution
$$x = - \frac{c}{b}$$
The graph
Rapid solution
[src]
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\| | / 2 / / 2 2 \\| | | / 2 / / 2 2 \\| | / 2 / / 2 2 \\|
|| 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/|| | 4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/||
||-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*re(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*im(a)| |-im(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------||*im(a) |-re(b) + \/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------||*re(a)
|\ \ 2 // \ \ 2 // | \ \ 2 // \ \ 2 //
x1 = I*|--------------------------------------------------------------------------------------------------------------------------------------------------------- - ---------------------------------------------------------------------------------------------------------------------------------------------------------| + --------------------------------------------------------------------------------------------------------------------------------------------------------- + ---------------------------------------------------------------------------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{1} = \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(\frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} - \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
// ________________________________________________________________ \ / ________________________________________________________________ \ \ / ________________________________________________________________ \ / ________________________________________________________________ \
|| / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ | | | / 2 / / 2 2 \\ | | / 2 / / 2 2 \\ |
||4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| | |4 / 2 / 2 2 \ |atan2\-4*im(a*c) + 2*im(b)*re(b), re (b) - im (b) - 4*re(a*c)/| |
||\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*im(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*re(a)| |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *cos|--------------------------------------------------------------| + re(b)|*re(a) |\/ (-4*im(a*c) + 2*im(b)*re(b)) + \re (b) - im (b) - 4*re(a*c)/ *sin|--------------------------------------------------------------| + im(b)|*im(a)
|\ \ 2 / / \ \ 2 / / | \ \ 2 / / \ \ 2 / /
x2 = I*|-------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------| - -------------------------------------------------------------------------------------------------------------------------------------------------------- - --------------------------------------------------------------------------------------------------------------------------------------------------------
| / 2 2 \ / 2 2 \ | / 2 2 \ / 2 2 \
\ 2*\im (a) + re (a)/ 2*\im (a) + re (a)/ / 2*\im (a) + re (a)/ 2*\im (a) + re (a)/
$$x_{2} = - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} - \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + i \left(- \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{im}{\left(b\right)}\right) \operatorname{re}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)} + \frac{\left(\sqrt[4]{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)}\right)^{2} + \left(\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(b\right)} \operatorname{im}{\left(b\right)} - 4 \operatorname{im}{\left(a c\right)},\left(\operatorname{re}{\left(b\right)}\right)^{2} - 4 \operatorname{re}{\left(a c\right)} - \left(\operatorname{im}{\left(b\right)}\right)^{2} \right)}}{2} \right)} + \operatorname{re}{\left(b\right)}\right) \operatorname{im}{\left(a\right)}}{2 \left(\left(\operatorname{re}{\left(a\right)}\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}\right)}\right)$$
x2 = -(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c, re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*im(a)/(2*(re(a)^2 + im(a)^2)) - (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + i*(-(((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*sin(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + im(b))*re(a)/(2*(re(a)^2 + im(a)^2)) + (((2*re(b)*im(b) - 4*im(a*c))^2 + (re(b)^2 - 4*re(a*c) - im(b)^2)^2)^(1/4)*cos(atan2(2*re(b)*im(b) - 4*im(a*c), re(b)^2 - 4*re(a*c) - im(b)^2)/2) + re(b))*im(a)/(2*(re(a)^2 + im(a)^2))))