Mister Exam
((4^x)+(2^x-3))^2+28((4^x)+(2^x-3))+192=>0 inequation
The solution
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2 / x x \ / x x \ \4 + 2 - 3/ + 28*\4 + 2 - 3/ + 192 >= 0
$$\left(\left(4^{x} + \left(2^{x} - 3\right)\right)^{2} + 28 \left(4^{x} + \left(2^{x} - 3\right)\right)\right) + 192 \geq 0$$
(4^x + 2^x - 3)^2 + 28*(4^x + 2^x - 3) + 192 >= 0
Detail solution
Given the inequality:
$$\left(\left(4^{x} + \left(2^{x} - 3\right)\right)^{2} + 28 \left(4^{x} + \left(2^{x} - 3\right)\right)\right) + 192 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(4^{x} + \left(2^{x} - 3\right)\right)^{2} + 28 \left(4^{x} + \left(2^{x} - 3\right)\right)\right) + 192 = 0$$
Solve:
$$x_{1} = \frac{\log{\left(- \frac{1}{2} - \frac{\sqrt{35} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{35} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{3} = \frac{\log{\left(- \frac{1}{2} - \frac{\sqrt{51} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{4} = \frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{51} i}{2} \right)}}{\log{\left(2 \right)}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
$$\left(28 \left(\left(-3 + 2^{0}\right) + 4^{0}\right) + \left(\left(-3 + 2^{0}\right) + 4^{0}\right)^{2}\right) + 192 \geq 0$$
so the inequality is always executed
$$\left(\left(4^{x} + \left(2^{x} - 3\right)\right)^{2} + 28 \left(4^{x} + \left(2^{x} - 3\right)\right)\right) + 192 \geq 0$$
To solve this inequality, we must first solve the corresponding equation:
$$\left(\left(4^{x} + \left(2^{x} - 3\right)\right)^{2} + 28 \left(4^{x} + \left(2^{x} - 3\right)\right)\right) + 192 = 0$$
Solve:
$$x_{1} = \frac{\log{\left(- \frac{1}{2} - \frac{\sqrt{35} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{2} = \frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{35} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{3} = \frac{\log{\left(- \frac{1}{2} - \frac{\sqrt{51} i}{2} \right)}}{\log{\left(2 \right)}}$$
$$x_{4} = \frac{\log{\left(- \frac{1}{2} + \frac{\sqrt{51} i}{2} \right)}}{\log{\left(2 \right)}}$$
Exclude the complex solutions:
This equation has no roots,
this inequality is executed for any x value or has no solutions
check it
subtitute random point x, for example
x0 = 0
$$\left(28 \left(\left(-3 + 2^{0}\right) + 4^{0}\right) + \left(\left(-3 + 2^{0}\right) + 4^{0}\right)^{2}\right) + 192 \geq 0$$
165 >= 0
so the inequality is always executed
Rapid solution
This inequality holds true always