Mister Exam
Other calculators
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How to use it?
- Equation:
-
Equation x^2-7*x+10=0
-
Equation x^2-18=7*x
-
Equation x^2+9=0
-
Equation -7x=4
- Express {x} in terms of y where:
- 17*x-19*y=-12
- 13*x-1*y=-20
- -6*x-11*y=-20
- -15*x+7*y=1
- Integral of d{x}:
- ax^2
- Derivative of:
- ax^2
-
Identical expressions
- ax^ two =c
- ax squared equally c
- ax to the power of two equally c
- ax2=c
- ax²=c
- ax to the power of 2=c
ax^2=c equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
Move right part of the equation to
left part with negative sign.
The equation is transformed from
$$a x^{2} = c$$
to
$$a x^{2} - c = 0$$
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
$$b = 0$$
$$c = - c$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{a c}}{a}$$
$$x_{2} = - \frac{\sqrt{a c}}{a}$$
left part with negative sign.
The equation is transformed from
$$a x^{2} = c$$
to
$$a x^{2} - c = 0$$
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
True
$$b = 0$$
$$c = - c$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (a) * (-c) = 4*a*c
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{a c}}{a}$$
$$x_{2} = - \frac{\sqrt{a c}}{a}$$
The solution of the parametric equation
Given the equation with a parameter:
$$a x^{2} = c$$
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
$$- c - x^{2} = 0$$
its solution
$$x = - \sqrt{- c}$$
$$x = \sqrt{- c}$$
With
$$a = 0$$
the equation
$$- c = 0$$
its solution
$$a x^{2} = c$$
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
$$- c - x^{2} = 0$$
its solution
$$x = - \sqrt{- c}$$
$$x = \sqrt{- c}$$
With
$$a = 0$$
the equation
$$- c = 0$$
its solution
Vieta's Theorem
rewrite the equation
$$a x^{2} = c$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - c}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{c}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{c}{a}$$
$$a x^{2} = c$$
of
$$a x^{2} + b x + c = 0$$
as reduced quadratic equation
$$x^{2} + \frac{b x}{a} + \frac{c}{a} = 0$$
$$\frac{a x^{2} - c}{a} = 0$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{c}{a}$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{c}{a}$$
The graph
Sum and product of roots
[src]
sum
/ / /c\ /c\\\ / / /c\ /c\\\ / / /c\ /c\\\ / / /c\ /c\\\
_________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-|||
/ 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//|
- 4 / im |-| + re |-| *cos|-------------------| - I*4 / im |-| + re |-| *sin|-------------------| + 4 / im |-| + re |-| *cos|-------------------| + I*4 / im |-| + re |-| *sin|-------------------|
\/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 /
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right)$$
=
0
$$0$$
product
/ / / /c\ /c\\\ / / /c\ /c\\\\ / / / /c\ /c\\\ / / /c\ /c\\\\ | _________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-|||| | _________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-|||| | / 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//|| | / 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//|| |- 4 / im |-| + re |-| *cos|-------------------| - I*4 / im |-| + re |-| *sin|-------------------||*|4 / im |-| + re |-| *cos|-------------------| + I*4 / im |-| + re |-| *sin|-------------------|| \ \/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 // \\/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 //
$$\left(- i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}\right)$$
=
/ /c\ /c\\
_________________ I*atan2|im|-|, re|-||
/ 2/c\ 2/c\ \ \a/ \a//
- / im |-| + re |-| *e
\/ \a/ \a/
$$- \sqrt{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}$$
-sqrt(im(c/a)^2 + re(c/a)^2)*exp(i*atan2(im(c/a), re(c/a)))
Rapid solution
[src]
/ / /c\ /c\\\ / / /c\ /c\\\
_________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-|||
/ 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//|
x1 = - 4 / im |-| + re |-| *cos|-------------------| - I*4 / im |-| + re |-| *sin|-------------------|
\/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 /
$$x_{1} = - i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} - \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}$$
/ / /c\ /c\\\ / / /c\ /c\\\
_________________ |atan2|im|-|, re|-||| _________________ |atan2|im|-|, re|-|||
/ 2/c\ 2/c\ | \ \a/ \a//| / 2/c\ 2/c\ | \ \a/ \a//|
x2 = 4 / im |-| + re |-| *cos|-------------------| + I*4 / im |-| + re |-| *sin|-------------------|
\/ \a/ \a/ \ 2 / \/ \a/ \a/ \ 2 /
$$x_{2} = i \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)} + \sqrt[4]{\left(\operatorname{re}{\left(\frac{c}{a}\right)}\right)^{2} + \left(\operatorname{im}{\left(\frac{c}{a}\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(\operatorname{im}{\left(\frac{c}{a}\right)},\operatorname{re}{\left(\frac{c}{a}\right)} \right)}}{2} \right)}$$
x2 = i*(re(c/a)^2 + im(c/a)^2)^(1/4)*sin(atan2(im(c/a, re(c/a))/2) + (re(c/a)^2 + im(c/a)^2)^(1/4)*cos(atan2(im(c/a), re(c/a))/2))