Mister Exam
|(x+1)3|=X3 equation
The teacher will be very surprised to see your correct solution 😉
The solution
Detail solution
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$3 x + 3 \geq 0$$
or
$$-1 \leq x \wedge x < \infty$$
we get the equation
$$- x_{3} + \left(3 x + 3\right) = 0$$
after simplifying we get
$$3 x - x_{3} + 3 = 0$$
the solution in this interval:
$$x_{1} = \frac{x_{3}}{3} - 1$$
2.
$$3 x + 3 < 0$$
or
$$-\infty < x \wedge x < -1$$
we get the equation
$$- x_{3} + \left(- 3 x - 3\right) = 0$$
after simplifying we get
$$- 3 x - x_{3} - 3 = 0$$
the solution in this interval:
$$x_{2} = - \frac{x_{3}}{3} - 1$$
The final answer:
$$x_{1} = \frac{x_{3}}{3} - 1$$
$$x_{2} = - \frac{x_{3}}{3} - 1$$
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$3 x + 3 \geq 0$$
or
$$-1 \leq x \wedge x < \infty$$
we get the equation
$$- x_{3} + \left(3 x + 3\right) = 0$$
after simplifying we get
$$3 x - x_{3} + 3 = 0$$
the solution in this interval:
$$x_{1} = \frac{x_{3}}{3} - 1$$
2.
$$3 x + 3 < 0$$
or
$$-\infty < x \wedge x < -1$$
we get the equation
$$- x_{3} + \left(- 3 x - 3\right) = 0$$
after simplifying we get
$$- 3 x - x_{3} - 3 = 0$$
the solution in this interval:
$$x_{2} = - \frac{x_{3}}{3} - 1$$
The final answer:
$$x_{1} = \frac{x_{3}}{3} - 1$$
$$x_{2} = - \frac{x_{3}}{3} - 1$$
The graph
Rapid solution
[src]
// x3 \ // x3 \
||-1 - -- for x3 > 0| ||-1 - -- for x3 > 0|
x1 = I*im|< 3 | + re|< 3 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// x3 \ // x3 \
||-1 + -- for x3 >= 0| ||-1 + -- for x3 >= 0|
x2 = I*im|< 3 | + re|< 3 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{2} = \operatorname{re}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
Eq(x2, re(Piecewise((x3/3 - 1, x3 >= 0), (nan, True))) + i*im(Piecewise((x3/3 - 1, x3 >= 0), (nan, True))))
Sum and product of roots
[src]
sum
// x3 \ // x3 \ // x3 \ // x3 \
||-1 - -- for x3 > 0| ||-1 - -- for x3 > 0| ||-1 + -- for x3 >= 0| ||-1 + -- for x3 >= 0|
I*im|< 3 | + re|< 3 | + I*im|< 3 | + re|< 3 |
|| | || | || | || |
\\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise /
$$\left(\operatorname{re}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) + \left(\operatorname{re}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
// x3 \ // x3 \ // x3 \ // x3 \
||-1 - -- for x3 > 0| ||-1 + -- for x3 >= 0| ||-1 - -- for x3 > 0| ||-1 + -- for x3 >= 0|
I*im|< 3 | + I*im|< 3 | + re|< 3 | + re|< 3 |
|| | || | || | || |
\\ nan otherwise / \\ nan otherwise / \\ nan otherwise / \\ nan otherwise /
$$\operatorname{re}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + \operatorname{re}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
product
/ // x3 \ // x3 \\ / // x3 \ // x3 \\ | ||-1 - -- for x3 > 0| ||-1 - -- for x3 > 0|| | ||-1 + -- for x3 >= 0| ||-1 + -- for x3 >= 0|| |I*im|< 3 | + re|< 3 ||*|I*im|< 3 | + re|< 3 || | || | || || | || | || || \ \\ nan otherwise / \\ nan otherwise // \ \\ nan otherwise / \\ nan otherwise //
$$\left(\operatorname{re}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right) \left(\operatorname{re}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{x_{3}}{3} - 1 & \text{for}\: x_{3} \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}\right)$$
=
/-(-3 + I*im(x3) + re(x3))*(3 + I*im(x3) + re(x3)) |-------------------------------------------------- for x3 > 0 < 9 | \ nan otherwise
$$\begin{cases} - \frac{\left(\operatorname{re}{\left(x_{3}\right)} + i \operatorname{im}{\left(x_{3}\right)} - 3\right) \left(\operatorname{re}{\left(x_{3}\right)} + i \operatorname{im}{\left(x_{3}\right)} + 3\right)}{9} & \text{for}\: x_{3} > 0 \\\text{NaN} & \text{otherwise} \end{cases}$$
Piecewise((-(-3 + i*im(x3) + re(x3))*(3 + i*im(x3) + re(x3))/9, x3 > 0), (nan, True))