Mister Exam
Other calculators
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How to use it?
- Equation:
- Equation a*x=b
-
Equation 3x^2-2x=0
- Equation x^2+y^2=1
-
Equation x^2-8x+16=0
- Express {x} in terms of y where:
- -19*x+16*y=-9
- -1*x-20*y=1
- -5*x+1*y=19
- -10*x+7*y=-12
- Canonical form:
-
x^2-y^2
- Factor polynomial:
- x^2-y^2
- Inequation:
- x^2-y^2
-
Identical expressions
- x^ two -y^ two = four
- x squared minus y squared equally 4
- x to the power of two minus y to the power of two equally four
- x2-y2=4
- x²-y²=4
- x to the power of 2-y to the power of 2=4
-
Similar expressions
- x^2+y^2=4
x^2-y^2=4 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
$$x^{2} - y^{2} = 4$$
to
$$\left(x^{2} - y^{2}\right) - 4 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - y^{2} - 4$$
, then
The equation has two roots.
or
$$x_{1} = \frac{\sqrt{4 y^{2} + 16}}{2}$$
$$x_{2} = - \frac{\sqrt{4 y^{2} + 16}}{2}$$
left part with negative sign.
The equation is transformed from
$$x^{2} - y^{2} = 4$$
to
$$\left(x^{2} - y^{2}\right) - 4 = 0$$
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
$$a = 1$$
$$b = 0$$
$$c = - y^{2} - 4$$
, then
D = b^2 - 4 * a * c =
(0)^2 - 4 * (1) * (-4 - y^2) = 16 + 4*y^2
The equation has two roots.
x1 = (-b + sqrt(D)) / (2*a)
x2 = (-b - sqrt(D)) / (2*a)
or
$$x_{1} = \frac{\sqrt{4 y^{2} + 16}}{2}$$
$$x_{2} = - \frac{\sqrt{4 y^{2} + 16}}{2}$$
Vieta's Theorem
it is reduced quadratic equation
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - y^{2} - 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - y^{2} - 4$$
$$p x + q + x^{2} = 0$$
where
$$p = \frac{b}{a}$$
$$p = 0$$
$$q = \frac{c}{a}$$
$$q = - y^{2} - 4$$
Vieta Formulas
$$x_{1} + x_{2} = - p$$
$$x_{1} x_{2} = q$$
$$x_{1} + x_{2} = 0$$
$$x_{1} x_{2} = - y^{2} - 4$$
The graph
Rapid solution
[src]
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/|
x1 = - \/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| - I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
\ 2 / \ 2 /
$$x_{1} = - i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}$$
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/|
x2 = \/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
\ 2 / \ 2 /
$$x_{2} = i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}$$
x2 = i*((re(y)^2 - im(y)^2 + 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*sin(atan2(2*re(y)*im(y, re(y)^2 - im(y)^2 + 4)/2) + ((re(y)^2 - im(y)^2 + 4)^2 + 4*re(y)^2*im(y)^2)^(1/4)*cos(atan2(2*re(y)*im(y), re(y)^2 - im(y)^2 + 4)/2))
Sum and product of roots
[src]
sum
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/ 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\ / 2 / / 2 2 \\
4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/|
- \/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| - I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------| + \/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|
\ 2 / \ 2 / \ 2 / \ 2 /
$$\left(- i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) + \left(i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right)$$
=
0
$$0$$
product
/ __________________________________________ __________________________________________ \ / __________________________________________ __________________________________________ \ | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | / 2 / / 2 2 \\ / 2 / / 2 2 \\| | 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/|| |4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/| 4 / / 2 2 \ 2 2 |atan2\2*im(y)*re(y), 4 + re (y) - im (y)/|| |- \/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| - I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------||*|\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *cos|-----------------------------------------| + I*\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *sin|-----------------------------------------|| \ \ 2 / \ 2 // \ \ 2 / \ 2 //
$$\left(- i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} - \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right) \left(i \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \sin{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)} + \sqrt[4]{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} \cos{\left(\frac{\operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}{2} \right)}\right)$$
=
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/ 2 / 2 2 \
/ / 2 2 \ 2 2 I*atan2\2*im(y)*re(y), 4 + re (y) - im (y)/
-\/ \4 + re (y) - im (y)/ + 4*im (y)*re (y) *e
$$- \sqrt{\left(\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4\right)^{2} + 4 \left(\operatorname{re}{\left(y\right)}\right)^{2} \left(\operatorname{im}{\left(y\right)}\right)^{2}} e^{i \operatorname{atan_{2}}{\left(2 \operatorname{re}{\left(y\right)} \operatorname{im}{\left(y\right)},\left(\operatorname{re}{\left(y\right)}\right)^{2} - \left(\operatorname{im}{\left(y\right)}\right)^{2} + 4 \right)}}$$
-sqrt((4 + re(y)^2 - im(y)^2)^2 + 4*im(y)^2*re(y)^2)*exp(i*atan2(2*im(y)*re(y), 4 + re(y)^2 - im(y)^2))