Mister Exam
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3*x^2-24*x+64=a*absolute(x-3) equation
The teacher will be very surprised to see your correct solution 😉
The solution
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[src]
2 3*x - 24*x + 64 = a*|x - 3|
$$\left(3 x^{2} - 24 x\right) + 64 = a \left|{x - 3}\right|$$
Detail solution
For every modulo expressions in the equation
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$x - 3 \geq 0$$
or
$$3 \leq x \wedge x < \infty$$
we get the equation
$$- a \left(x - 3\right) + 3 x^{2} - 24 x + 64 = 0$$
after simplifying we get
$$- a \left(x - 3\right) + 3 x^{2} - 24 x + 64 = 0$$
the solution in this interval:
$$x_{1} = \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{2} = \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
2.
$$x - 3 < 0$$
or
$$-\infty < x \wedge x < 3$$
we get the equation
$$- a \left(3 - x\right) + 3 x^{2} - 24 x + 64 = 0$$
after simplifying we get
$$- a \left(3 - x\right) + 3 x^{2} - 24 x + 64 = 0$$
the solution in this interval:
$$x_{3} = - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
$$x_{4} = - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
The final answer:
$$x_{1} = \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{2} = \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{3} = - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
$$x_{4} = - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
allow cases, when this expressions ">=0" or "<0",
solve the resulting equation.
1.
$$x - 3 \geq 0$$
or
$$3 \leq x \wedge x < \infty$$
we get the equation
$$- a \left(x - 3\right) + 3 x^{2} - 24 x + 64 = 0$$
after simplifying we get
$$- a \left(x - 3\right) + 3 x^{2} - 24 x + 64 = 0$$
the solution in this interval:
$$x_{1} = \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{2} = \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
2.
$$x - 3 < 0$$
or
$$-\infty < x \wedge x < 3$$
we get the equation
$$- a \left(3 - x\right) + 3 x^{2} - 24 x + 64 = 0$$
after simplifying we get
$$- a \left(3 - x\right) + 3 x^{2} - 24 x + 64 = 0$$
the solution in this interval:
$$x_{3} = - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
$$x_{4} = - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
The final answer:
$$x_{1} = \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{2} = \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4$$
$$x_{3} = - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
$$x_{4} = - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4$$
The graph
Rapid solution
[src]
// __________________ __________________ \ // __________________ __________________ \
|| / 2 / 2 | || / 2 / 2 |
|| a \/ -192 + a - 12*a a \/ -192 + a - 12*a | || a \/ -192 + a - 12*a a \/ -192 + a - 12*a |
x1 = I*im|<4 - - - --------------------- for -1 + - + --------------------- > 0| + re|<4 - - - --------------------- for -1 + - + --------------------- > 0|
|| 6 6 6 6 | || 6 6 6 6 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{1} = \operatorname{re}{\left(\begin{cases} - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} - 1 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{6} - \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} - 1 > 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// __________________ __________________ \ // __________________ __________________ \
|| / 2 / 2 | || / 2 / 2 |
|| a \/ -192 + a - 12*a a \/ -192 + a - 12*a | || a \/ -192 + a - 12*a a \/ -192 + a - 12*a |
x2 = I*im|<4 - - + --------------------- for 1 - - + --------------------- < 0| + re|<4 - - + --------------------- for 1 - - + --------------------- < 0|
|| 6 6 6 6 | || 6 6 6 6 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{2} = \operatorname{re}{\left(\begin{cases} - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4 & \text{for}\: - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 1 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 4 & \text{for}\: - \frac{a}{6} + \frac{\sqrt{a^{2} - 12 a - 192}}{6} + 1 < 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// __________________ __________________ \ // __________________ __________________ \
|| / 2 / 2 | || / 2 / 2 |
|| \/ -192 + a + 12*a a \/ -192 + a + 12*a a | || \/ -192 + a + 12*a a \/ -192 + a + 12*a a |
x3 = I*im|<4 - --------------------- + - for 1 - --------------------- + - >= 0| + re|<4 - --------------------- + - for 1 - --------------------- + - >= 0|
|| 6 6 6 6 | || 6 6 6 6 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{3} = \operatorname{re}{\left(\begin{cases} \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 1 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} - \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 1 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
// __________________ __________________ \ // __________________ __________________ \
|| / 2 / 2 | || / 2 / 2 |
|| a \/ -192 + a + 12*a a \/ -192 + a + 12*a | || a \/ -192 + a + 12*a a \/ -192 + a + 12*a |
x4 = I*im|<4 + - + --------------------- for 1 + - + --------------------- >= 0| + re|<4 + - + --------------------- for 1 + - + --------------------- >= 0|
|| 6 6 6 6 | || 6 6 6 6 |
|| | || |
\\ nan otherwise / \\ nan otherwise /
$$x_{4} = \operatorname{re}{\left(\begin{cases} \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 1 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)} + i \operatorname{im}{\left(\begin{cases} \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 4 & \text{for}\: \frac{a}{6} + \frac{\sqrt{a^{2} + 12 a - 192}}{6} + 1 \geq 0 \\\text{NaN} & \text{otherwise} \end{cases}\right)}$$
x4 = re(Piecewise((a/6 + sqrt(a^2 + 12*a - 192/6 + 4, a/6 + sqrt(a^2 + 12*a - 192)/6 + 1 >= 0), (nan, True))) + i*im(Piecewise((a/6 + sqrt(a^2 + 12*a - 192)/6 + 4, a/6 + sqrt(a^2 + 12*a - 192)/6 + 1 >= 0), (nan, True))))