Sr Examen
Otras calculadoras
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¿Cómo usar?
- La ecuación:
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Ecuación x^4+2*x^2-8=0
-
Ecuación 1/(x-3)^2-3/(x-3)-4=0
-
Ecuación 9*(x-5)=-x
-
Ecuación 13x+10=6x-4
- Expresar {x} en función de y en la ecuación:
- 7*x-6*y=-2
- 10*x-2*y=0
- 13*x-15*y=-6
- 4*x-7*y=4
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Expresiones idénticas
- absolute(x^ dos - dos)=a
- absolute(x al cuadrado menos 2) es igual a a
- absolute(x en el grado dos menos dos) es igual a a
- absolute(x2-2)=a
- absolutex2-2=a
- absolute(x²-2)=a
- absolute(x en el grado 2-2)=a
- absolutex^2-2=a
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Expresiones semejantes
- absolute(x^2+2)=a
-
Expresiones con funciones
- absolute
- absolute(x^2-9x+7)=7
- absolute(x-1)=0/(1-x)
- absolute(sin(x))+absolute(cos(x))=1
- absolute(z)-z=1+2*i
- absolute(x+3)-absolute(x-2)=5
absolute(x^2-2)=a la ecuación
El profesor se sorprenderá mucho al ver tu solución correcta😉
Solución
Solución detallada
Para cada expresión dentro del módulo en la ecuación
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.
1.
$$x^{2} - 2 \geq 0$$
o
$$\left(x \leq - \sqrt{2} \wedge -\infty < x\right) \vee \left(\sqrt{2} \leq x \wedge x < \infty\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 2\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 2 = 0$$
la resolución en este intervalo:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
2.
$$x^{2} - 2 < 0$$
o
$$- \sqrt{2} < x \wedge x < \sqrt{2}$$
obtenemos la ecuación
$$- a + \left(2 - x^{2}\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 2 = 0$$
la resolución en este intervalo:
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.
1.
$$x^{2} - 2 \geq 0$$
o
$$\left(x \leq - \sqrt{2} \wedge -\infty < x\right) \vee \left(\sqrt{2} \leq x \wedge x < \infty\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 2\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 2 = 0$$
la resolución en este intervalo:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
2.
$$x^{2} - 2 < 0$$
o
$$- \sqrt{2} < x \wedge x < \sqrt{2}$$
obtenemos la ecuación
$$- a + \left(2 - x^{2}\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 2 = 0$$
la resolución en este intervalo:
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
Entonces la respuesta definitiva es:
$$x_{1} = - \sqrt{a + 2}$$
$$x_{2} = \sqrt{a + 2}$$
$$x_{3} = - \sqrt{2 - a}$$
$$x_{4} = \sqrt{2 - a}$$
Gráfica
Suma y producto de raíces
[src]
suma
// _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \
||-\/ 2 - a for a > 0| ||-\/ 2 - a for a > 0| ||\/ 2 - a for a > 0| ||\/ 2 - a for a > 0| ||-\/ 2 + a for a >= 0| ||-\/ 2 + a for a >= 0| ||\/ 2 + a for a >= 0| ||\/ 2 + a for a >= 0|
I*im|< | + re|< | + I*im|< | + re|< | + I*im|< | + re|< | + I*im|< | + re|< |
|| nan otherwise| || nan otherwise| || nan otherwise| || nan otherwise| || nan otherwise | || nan otherwise | || nan otherwise | || nan otherwise |
\\ / \\ / \\ / \\ / \\ / \\ / \\ / \\ /
$$\left(\left(\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
// _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \ // _______ \
||\/ 2 + a for a >= 0| ||\/ 2 - a for a > 0| ||-\/ 2 + a for a >= 0| ||-\/ 2 - a for a > 0| ||\/ 2 + a for a >= 0| ||\/ 2 - a for a > 0| ||-\/ 2 + a for a >= 0| ||-\/ 2 - a for a > 0|
I*im|< | + I*im|< | + I*im|< | + I*im|< | + re|< | + re|< | + re|< | + re|< |
|| nan otherwise | || nan otherwise| || nan otherwise | || nan otherwise| || nan otherwise | || nan otherwise| || nan otherwise | || nan otherwise|
\\ / \\ / \\ / \\ / \\ / \\ / \\ / \\ /
$$\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
producto
/ // _______ \ // _______ \\ / // _______ \ // _______ \\ / // _______ \ // _______ \\ / // _______ \ // _______ \\ | ||-\/ 2 - a for a > 0| ||-\/ 2 - a for a > 0|| | ||\/ 2 - a for a > 0| ||\/ 2 - a for a > 0|| | ||-\/ 2 + a for a >= 0| ||-\/ 2 + a for a >= 0|| | ||\/ 2 + a for a >= 0| ||\/ 2 + a for a >= 0|| |I*im|< | + re|< ||*|I*im|< | + re|< ||*|I*im|< | + re|< ||*|I*im|< | + re|< || | || nan otherwise| || nan otherwise|| | || nan otherwise| || nan otherwise|| | || nan otherwise | || nan otherwise || | || nan otherwise | || nan otherwise || \ \\ / \\ // \ \\ / \\ // \ \\ / \\ // \ \\ / \\ //
$$\left(\operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
/ ________________________ _______________________ | / 2 2 / 2 2 I*(atan2(-im(a), 2 - re(a)) + atan2(im(a), 2 + re(a))) <\/ (-2 + re(a)) + im (a) *\/ (2 + re(a)) + im (a) *e for a > 0 | \ nan otherwise
$$\begin{cases} \sqrt{\left(\operatorname{re}{\left(a\right)} - 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} \sqrt{\left(\operatorname{re}{\left(a\right)} + 2\right)^{2} + \left(\operatorname{im}{\left(a\right)}\right)^{2}} e^{i \left(\operatorname{atan_{2}}{\left(- \operatorname{im}{\left(a\right)},2 - \operatorname{re}{\left(a\right)} \right)} + \operatorname{atan_{2}}{\left(\operatorname{im}{\left(a\right)},\operatorname{re}{\left(a\right)} + 2 \right)}\right)} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
Piecewise((sqrt((-2 + re(a))^2 + im(a)^2)*sqrt((2 + re(a))^2 + im(a)^2)*exp(i*(atan2(-im(a), 2 - re(a)) + atan2(im(a), 2 + re(a)))), a > 0), (nan, True))
Respuesta rápida
[src]
// _______ \ // _______ \
||-\/ 2 - a for a > 0| ||-\/ 2 - a for a > 0|
x1 = I*im|< | + re|< |
|| nan otherwise| || nan otherwise|
\\ / \\ /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||\/ 2 - a for a > 0| ||\/ 2 - a for a > 0|
x2 = I*im|< | + re|< |
|| nan otherwise| || nan otherwise|
\\ / \\ /
$$x_{2} = \operatorname{re}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{2 - a} & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||-\/ 2 + a for a >= 0| ||-\/ 2 + a for a >= 0|
x3 = I*im|< | + re|< |
|| nan otherwise | || nan otherwise |
\\ / \\ /
$$x_{3} = \operatorname{re}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// _______ \ // _______ \
||\/ 2 + a for a >= 0| ||\/ 2 + a for a >= 0|
x4 = I*im|< | + re|< |
|| nan otherwise | || nan otherwise |
\\ / \\ /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 2} & \text{for}\: a \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((sqrt(a + 2, a >= 0), (nan, True))) + i*im(Piecewise((sqrt(a + 2), a >= 0), (nan, True))))