Mister Exam
Other calculators
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How to use it?
- Equation:
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Equation (x-3)*(x+4)=0
-
Equation x^3-x^2-5=0
-
Equation x-2(3x+2)=16
- Equation x^2-1=1+(1/2)*y
- Express {x} in terms of y where:
- -10*x+16*y=-6
- -13*x+9*y=1
- -10*x+9*y=-6
- -17*x+13*y=-3
- Derivative of:
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x^2-1
- Inequation:
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x^2-1
- Graphing y =:
-
x^2-1
-
Identical expressions
- x^ two - one = one +(one / two)*y
- x squared minus 1 equally 1 plus (1 divide by 2) multiply by y
- x to the power of two minus one equally one plus (one divide by two) multiply by y
- x2-1=1+(1/2)*y
- x2-1=1+1/2*y
- x²-1=1+(1/2)*y
- x to the power of 2-1=1+(1/2)*y
- x^2-1=1+(1/2)y
- x2-1=1+(1/2)y
- x2-1=1+1/2y
- x^2-1=1+1/2y
- x^2-1=1+(1 divide by 2)*y
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Similar expressions
- x^3+x^2-12*x=0
- x^2-1000x+160000=0
- x^2+1=1+(1/2)*y
- √(x^2-1)=2*x
- x^2-1=1-(1/2)*y
- 2x^2-10x=0
x^2-1=1+(1/2)*y 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
$$x^{2} - 1 = \frac{y}{2} + 1$$
to
$$\left(x^{2} - 1\right) + \left(- \frac{y}{2} - 1\right) = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - \frac{y}{2} - 2$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{2 y + 8}}{2}$$
$$x_{2} = - \frac{\sqrt{2 y + 8}}{2}$$
left part with negative sign.
The equation is transformed from
$$x^{2} - 1 = \frac{y}{2} + 1$$
to
$$\left(x^{2} - 1\right) + \left(- \frac{y}{2} - 1\right) = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - \frac{y}{2} - 2$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-2 - y/2) = 8 + 2*y
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{2 y + 8}}{2}$$
$$x_{2} = - \frac{\sqrt{2 y + 8}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{y}{2} - 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{y}{2} - 2$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - \frac{y}{2} - 2$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - \frac{y}{2} - 2$$
The graph
Rapid solution
[src]
___________________________ ___________________________
4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\
\/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------|
\ 2 / \ 2 /
x1 = - --------------------------------------------------------------- - -----------------------------------------------------------------
2 2
$$x_{1} = - \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}$$
___________________________ ___________________________
4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\
\/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------|
\ 2 / \ 2 /
x2 = --------------------------------------------------------------- + -----------------------------------------------------------------
2 2
$$x_{2} = \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}$$
x2 = i*((2*re(y) + 8)^2 + 4*im(y)^2)^(1/4)*sin(atan2(2*im(y, 2*re(y) + 8)/2)/2 + ((2*re(y) + 8)^2 + 4*im(y)^2)^(1/4)*cos(atan2(2*im(y), 2*re(y) + 8)/2)/2)
Sum and product of roots
[src]
sum
___________________________ ___________________________ ___________________________ ___________________________
4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\
\/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------| \/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
- --------------------------------------------------------------- - ----------------------------------------------------------------- + --------------------------------------------------------------- + -----------------------------------------------------------------
2 2 2 2
$$\left(- \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) + \left(\frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right)$$
=
0
$$0$$
product
/ ___________________________ ___________________________ \ / ___________________________ ___________________________ \ | 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\| |4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\ 4 / 2 2 /atan2(2*im(y), 8 + 2*re(y))\| | \/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------|| |\/ (8 + 2*re(y)) + 4*im (y) *cos|---------------------------| I*\/ (8 + 2*re(y)) + 4*im (y) *sin|---------------------------|| | \ 2 / \ 2 /| | \ 2 / \ 2 /| |- --------------------------------------------------------------- - -----------------------------------------------------------------|*|--------------------------------------------------------------- + -----------------------------------------------------------------| \ 2 2 / \ 2 2 /
$$\left(- \frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} - \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right) \left(\frac{i \sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2} + \frac{\sqrt[4]{\left(2 \operatorname{re}{\left(y\right)} + 8\right)^{2} + 4 \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}{2} \right)}}{2}\right)$$
=
_______________________
/ 2 2 I*atan2(2*im(y), 8 + 2*re(y))
-\/ (4 + re(y)) + im (y) *e
-----------------------------------------------------------
2
$$- \frac{\sqrt{\left(\operatorname{re}{\left(y\right)} + 4\right)^{2} + \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{im}{\left(y\right)},2 \operatorname{re}{\left(y\right)} + 8 \right)}}}{2}$$
-sqrt((4 + re(y))^2 + im(y)^2)*exp(i*atan2(2*im(y), 8 + 2*re(y)))/2