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Question Number 176580 by mathlove last updated on 22/Sep/22

$$4^{x^2-2x+2}-2^{x^2-2x+3}+2=2^{x^2-2x+2}$$$$x=?$$

Answered by cortano1 last updated on 22/Sep/22

$${\left(2^{{\mathrm{x}}^2-2\mathrm{x}+2}\right)}^2-2.2^{{\mathrm{x}}^2-2\mathrm{x}+2}=2^{{\mathrm{x}}^2-2\mathrm{x}+2}$$$${\mathrm{y}}^2-3\mathrm{y}=0$$$$\mathrm{y}(\mathrm{y}-3)=0$$$$\mathrm{y}=0\left(\mathrm{rejected}\right)$$$$\mathrm{y}=3\Rightarrow 2^{{\mathrm{x}}^2-2\mathrm{x}+2}=3$$$$\Rightarrow {\mathrm{x}}^2-2\mathrm{x}+2=\log {}_2\left(3\right)$$$$\Rightarrow {(\mathrm{x}-1)}^2=\log {}_2\left(3\right)-1$$$$\Rightarrow \mathrm{x}=1\pm \sqrt{\log {}_2\left(\frac{3}{2}\right)}$$

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