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bearhunter [10]
3 years ago
15

Solve each equation (Polynomial equations)

Mathematics
1 answer:
lesya [120]3 years ago
8 0

{ {e}^{x} }^{2}  =  \frac{1}{ {e}^{2} } \times  {e}^{3x}  \\ \Leftrightarrow  { {e}^{x} }^{2}  =  {e}^{ 3x}  \times  {e}^{ - 2}  \\ \Leftrightarrow  { {e}^{x} }^{2}  =  {e}^{3x - 2}  \\ \Leftrightarrow  {x}^{2}  = 3x - 2 \\ \Leftrightarrow  {x}^{2}  - 3x + 2 = 0 \\ \Leftrightarrow (x - 1)(x - 2) = 0 \\ x = 1 \: \vee \: x = 2 \\  \\ {32}^{ {x}^{2} - 2x }  =  \frac{1}{ {4}^{x} } \\  \Leftrightarrow  {( {2}^{5} )}^{ {x}^{2} - 2x }  = {2}^{ - 2x}  \\ \Leftrightarrow  {2}^{5 {x}^{2} - 10x }  =  {2}^{ - 2x}  \\ \Leftrightarrow 5 {x}^{2}  - 10x =  - 2x \\ \Leftrightarrow 5 {x}^{2}    - 8x = 0 \\ \Leftrightarrow x(5x - 8) = 0 \\ \Leftrightarrow x = 0 \: \vee \: x =  \frac{8}{5}  \\  \\  { {e}^{x} }^{3}  =  {2}^{2x} \\ \Leftrightarrow  {x}^{3}  =  ln( {2}^{2x} )  \\ \Leftrightarrow  {x}^{3}  - 2x ln(2)  = 0 \\ \Leftrightarrow x( {x}^{2}  - 2 ln(2) ) = 0 \\ \Leftrightarrow x = 0 \:\vee \:  {x}^{2}  = 2 ln(2)  \\ \Leftrightarrow x = 0 \:\vee \: x =  \sqrt{2 ln(2) }  \:\vee \: x =  -  \sqrt{2 ln(2) }

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