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\( \left. \begin{array} { l l } { 1.2 .2 } & { 3 ^ { x } - 3 ^ { 2 - x } = 8 } \\ { 1.2 .3 } & { x - 4 x ^ { \frac { 1 } { 2 } } - 5 = 0 } \end{array} \right. \)

Ask by Coleman Burgess. in South Africa
Mar 14,2025

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Answer

The solutions are: - \( x = 2 \) for the first equation. - \( x = 25 \) for the second equation.

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Beyond the Answer

To solve the equation \( 3^{x} - 3^{2-x} = 8 \), first rewrite \( 3^{2-x} \) as \( \frac{9}{3^{x}} \). This transforms the equation into \( 3^x - \frac{9}{3^x} = 8 \). Multiply through by \( 3^x \) to clear the fraction, leading to \( (3^x)^2 - 8(3^x) - 9 = 0 \). Letting \( y = 3^x \), we have a quadratic equation, which can be solved using the quadratic formula. For the equation \( x - 4x^{\frac{1}{2}} - 5 = 0 \), substitute \( x^{\frac{1}{2}} = y \), making the equation \( y^2 - 4y - 5 = 0 \). Again, apply the quadratic formula to find the values of \( y \), and subsequently find \( x \) by squaring the solutions.

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