Question

Solve each equation (Polynomial equations)


`e^{x^(2) } =(1)/(e^(2) )` ·
`e^(3x)`


`32^{x^(2) -2x} = (1)/(4^(x) )`


`e^{x^(3) } = 2^(2x)`

Answer 1

`{ {e}^(x) }^(2) = \frac{1}{ {e}^(2) } * {e}^(3x) \\ \Leftrightarrow { {e}^(x) }^(2) = {e}^( 3x) * {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 = (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 = √(2 ln(2) ) \:\vee \: x = - √(2 ln(2) )`