Mister Exam
√(4^x-17*2^x+16)+√(4x-x^2)+√(x+3)≥2 inequation
The solution
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_________________ __________ / x x / 2 _______ \/ 4 - 17*2 + 16 + \/ 4*x - x + \/ x + 3 >= 2
$$\sqrt{x + 3} + \left(\sqrt{- x^{2} + 4 x} + \sqrt{\left(- 17 \cdot 2^{x} + 4^{x}\right) + 16}\right) \geq 2$$
sqrt(x + 3) + sqrt(-x^2 + 4*x) + sqrt(-17*2^x + 4^x + 16) >= 2
Detail solution
Given the inequality:
$$\sqrt{x + 3} + \left(\sqrt{- x^{2} + 4 x} + \sqrt{\left(- 17 \cdot 2^{x} + 4^{x}\right) + 16}\right) \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{x + 3} + \left(\sqrt{- x^{2} + 4 x} + \sqrt{\left(- 17 \cdot 2^{x} + 4^{x}\right) + 16}\right) = 2$$
Solve:
$$x_{1} = -0.00217552372643211 - 0.00450900918064287 i$$
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
$$\left(\sqrt{0 \cdot 4 - 0^{2}} + \sqrt{\left(- 17 \cdot 2^{0} + 4^{0}\right) + 16}\right) + \sqrt{3} \geq 2$$
but
so the inequality has no solutions
$$\sqrt{x + 3} + \left(\sqrt{- x^{2} + 4 x} + \sqrt{\left(- 17 \cdot 2^{x} + 4^{x}\right) + 16}\right) \geq 2$$
To solve this inequality, we must first solve the corresponding equation:
$$\sqrt{x + 3} + \left(\sqrt{- x^{2} + 4 x} + \sqrt{\left(- 17 \cdot 2^{x} + 4^{x}\right) + 16}\right) = 2$$
Solve:
$$x_{1} = -0.00217552372643211 - 0.00450900918064287 i$$
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
$$\left(\sqrt{0 \cdot 4 - 0^{2}} + \sqrt{\left(- 17 \cdot 2^{0} + 4^{0}\right) + 16}\right) + \sqrt{3} \geq 2$$
___
\/ 3 >= 2
but
___
\/ 3 < 2
so the inequality has no solutions
Solving inequality on a graph