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- La ecuación:
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Ecuación x^3-x^2-x-1=0
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Ecuación 12-4(x-3)=39-9x
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Ecuación 0,2x+2,7=1,4-1,1x
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Ecuación x^2=2x+8
- Expresar {x} en función de y en la ecuación:
- -11*x-14*y=-3
- -9*x-1*y=6
- -2*x+2*y=4
- -11*x-3*y=14
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Expresiones idénticas
- abs(x^ dos -2x- tres)=a
- abs(x al cuadrado menos 2x menos 3) es igual a a
- abs(x en el grado dos menos 2x menos tres) es igual a a
- abs(x2-2x-3)=a
- absx2-2x-3=a
- abs(x²-2x-3)=a
- abs(x en el grado 2-2x-3)=a
- absx^2-2x-3=a
-
Expresiones semejantes
- abs(x^2+2x-3)=a
- abs(x^2-2x+3)=a
-
Expresiones con funciones
- abs
- abs(z+1i)=abs(z+2)
- abs(cosx)+a=sin(3x)
- abs(x-1)-2*abs(x-2)+3*abs(x-3)=4
- absolutesinx=sinx+2cos(2x)
- abs(z)-iz=1-2i
abs(x^2-2x-3)=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.
$$- x^{2} + 2 x + 3 \geq 0$$
o
$$-1 \leq x \wedge x \leq 3$$
obtenemos la ecuación
$$- a + \left(- x^{2} + 2 x + 3\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 2 x + 3 = 0$$
la resolución en este intervalo:
$$x_{1} = 1 - \sqrt{4 - a}$$
$$x_{2} = \sqrt{4 - a} + 1$$
2.
$$- x^{2} + 2 x + 3 < 0$$
o
$$\left(-\infty < x \wedge x < -1\right) \vee \left(3 < x \wedge x < \infty\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 2 x - 3\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 2 x - 3 = 0$$
la resolución en este intervalo:
$$x_{3} = 1 - \sqrt{a + 4}$$
$$x_{4} = \sqrt{a + 4} + 1$$
Entonces la respuesta definitiva es:
$$x_{1} = 1 - \sqrt{4 - a}$$
$$x_{2} = \sqrt{4 - a} + 1$$
$$x_{3} = 1 - \sqrt{a + 4}$$
$$x_{4} = \sqrt{a + 4} + 1$$
admitimos los casos cuando la expresión correspondiente es ">= 0" o "< 0",
resolvemos las ecuaciones obtenidas.
1.
$$- x^{2} + 2 x + 3 \geq 0$$
o
$$-1 \leq x \wedge x \leq 3$$
obtenemos la ecuación
$$- a + \left(- x^{2} + 2 x + 3\right) = 0$$
simplificamos, obtenemos
$$- a - x^{2} + 2 x + 3 = 0$$
la resolución en este intervalo:
$$x_{1} = 1 - \sqrt{4 - a}$$
$$x_{2} = \sqrt{4 - a} + 1$$
2.
$$- x^{2} + 2 x + 3 < 0$$
o
$$\left(-\infty < x \wedge x < -1\right) \vee \left(3 < x \wedge x < \infty\right)$$
obtenemos la ecuación
$$- a + \left(x^{2} - 2 x - 3\right) = 0$$
simplificamos, obtenemos
$$- a + x^{2} - 2 x - 3 = 0$$
la resolución en este intervalo:
$$x_{3} = 1 - \sqrt{a + 4}$$
$$x_{4} = \sqrt{a + 4} + 1$$
Entonces la respuesta definitiva es:
$$x_{1} = 1 - \sqrt{4 - a}$$
$$x_{2} = \sqrt{4 - a} + 1$$
$$x_{3} = 1 - \sqrt{a + 4}$$
$$x_{4} = \sqrt{a + 4} + 1$$
Gráfica
Suma y producto de raíces
[src]
suma
// 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ |
I*im|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0| + re|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0| + I*im|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + re|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + I*im|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + re|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + I*im|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0| + re|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0|
|| | || | || | || | || | || | || | || |
\\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise /
$$\left(\left(\left(\operatorname{re}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)\right) + \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
// 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ | || _______ / _______\ _______ |
I*im|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0| + I*im|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + I*im|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + I*im|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0| + re|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0| + re|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + re|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + re|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0|
|| | || | || | || | || | || | || | || |
\\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise /
$$\operatorname{re}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
producto
/ // 2 \ // 2 \\ / // 2 \ // 2 \\ / // 2 \ // 2 \\ / // 2 \ // 2 \\ | || _______ / _______\ _______ | || _______ / _______\ _______ || | || _______ / _______\ _______ | || _______ / _______\ _______ || | || _______ / _______\ _______ | || _______ / _______\ _______ || | || _______ / _______\ _______ | || _______ / _______\ _______ || |I*im|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0| + re|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0||*|I*im|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + re|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0||*|I*im|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + re|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0||*|I*im|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0| + re|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0|| | || | || || | || | || || | || | || || | || | || || \ \\ nan otherwise / \\ nan otherwise // \ \\ nan otherwise / \\ nan otherwise // \ \\ nan otherwise / \\ nan otherwise // \ \\ nan otherwise / \\ nan otherwise //
$$\left(\operatorname{re}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
/ 2 2 |9 + im (a) - re (a) - 2*I*im(a)*re(a) for a > 0 < | nan otherwise \
$$\begin{cases} - \left(\operatorname{re}{\left(a\right)}\right)^{2} - 2 i \operatorname{re}{\left(a\right)} \operatorname{im}{\left(a\right)} + \left(\operatorname{im}{\left(a\right)}\right)^{2} + 9 & \text{for}\: a > 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
Piecewise((9 + im(a)^2 - re(a)^2 - 2*i*im(a)*re(a), a > 0), (nan, True))
Respuesta rápida
[src]
// 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ |
x1 = I*im|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0| + re|<1 - \/ 4 - a for -5 + \-1 + \/ 4 - a / + 2*\/ 4 - a <= 0|
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{1} = \operatorname{re}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{4 - a} & \text{for}\: 2 \sqrt{4 - a} + \left(\sqrt{4 - a} - 1\right)^{2} - 5 \leq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ |
x2 = I*im|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0| + re|<1 - \/ 4 + a for -5 + \-1 + \/ 4 + a / + 2*\/ 4 + a > 0|
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{2} = \operatorname{re}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} 1 - \sqrt{a + 4} & \text{for}\: 2 \sqrt{a + 4} + \left(\sqrt{a + 4} - 1\right)^{2} - 5 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ |
x3 = I*im|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0| + re|<1 + \/ 4 - a for 5 - \1 + \/ 4 - a / + 2*\/ 4 - a >= 0|
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{4 - a} + 1 & \text{for}\: 2 \sqrt{4 - a} - \left(\sqrt{4 - a} + 1\right)^{2} + 5 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// 2 \ // 2 \
|| _______ / _______\ _______ | || _______ / _______\ _______ |
x4 = I*im|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0| + re|<1 + \/ 4 + a for 5 - \1 + \/ 4 + a / + 2*\/ 4 + a < 0|
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \sqrt{a + 4} + 1 & \text{for}\: 2 \sqrt{a + 4} - \left(\sqrt{a + 4} + 1\right)^{2} + 5 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((sqrt(a + 4 + 1, 2*sqrt(a + 4) - (sqrt(a + 4) + 1)^2 + 5 < 0), (nan, True))) + i*im(Piecewise((sqrt(a + 4) + 1, 2*sqrt(a + 4) - (sqrt(a + 4) + 1)^2 + 5 < 0), (nan, True))))