Find the truth set of the following equatio a) \( 2^{2 x+1}-5\left(2^{x}\right)+2=0 \) b) \( 3^{2 x+3}-4\left(3^{x+1}\right)+1=0 \) c) \( 3^{2 x}-30\left(3^{x-2}\right)+1=0 \) d) \( 2^{2 x}+3\left(2^{x}\right)-4=0 \) e) \( 2^{2 x}-5\left(2^{x}\right)+2=0 \) \( 3^{2 x+1}-26\left(3^{x}\right)-9=0 \) e) \( 4^{x}-6\left(2^{x}\right)-16=0 \) \( 2^{x}+2^{(x-1)}=48 \)

Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x}+3\times 2^{x}-4=0\) - step1: Factor the expression: \(\left(2^{x}-1\right)\left(2^{x}+4\right)=0\) - step2: Separate into possible cases: \(\begin{align}&2^{x}-1=0\\&2^{x}+4=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=0\\&x \notin \mathbb{R}\end{align}\) - step4: Find the union: \(x=0\) Solve the equation \( 3^{2 x+3}-4(3^{x+1})+1=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x+3}-4\times 3^{x+1}+1=0\) - step1: Factor the expression: \(\left(3^{x+1}-1\right)\left(3^{x+2}-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&3^{x+1}-1=0\\&3^{x+2}-1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=-1\\&x=-2\end{align}\) - step4: Rewrite: \(x_{1}=-2,x_{2}=-1\) Solve the equation \( 3^{2 x+1}-26(3^{x})-9=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x+1}-26\times 3^{x}-9=0\) - step1: Factor the expression: \(\left(3^{x}-9\right)\left(3^{x+1}+1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&3^{x}-9=0\\&3^{x+1}+1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=2\\&x \notin \mathbb{R}\end{align}\) - step4: Find the union: \(x=2\) Solve the equation \( 4^{x}-6(2^{x})-16=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(4^{x}-6\times 2^{x}-16=0\) - step1: Factor the expression: \(\left(2^{x}-8\right)\left(2^{x}+2\right)=0\) - step2: Separate into possible cases: \(\begin{align}&2^{x}-8=0\\&2^{x}+2=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=3\\&x \notin \mathbb{R}\end{align}\) - step4: Find the union: \(x=3\) Solve the equation \( 2^{2 x+1}-5(2^{x})+2=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x+1}-5\times 2^{x}+2=0\) - step1: Factor the expression: \(\left(2^{x}-2\right)\left(2^{x+1}-1\right)=0\) - step2: Separate into possible cases: \(\begin{align}&2^{x}-2=0\\&2^{x+1}-1=0\end{align}\) - step3: Solve the equation: \(\begin{align}&x=1\\&x=-1\end{align}\) - step4: Rewrite: \(x_{1}=-1,x_{2}=1\) Solve the equation \( 2^{x}+2^{(x-1)}=48 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{x}+2^{\left(x-1\right)}=48\) - step1: Simplify: \(3\times 2^{x-1}=48\) - step2: Divide both sides: \(\frac{3\times 2^{x-1}}{3}=\frac{48}{3}\) - step3: Divide the numbers: \(2^{x-1}=16\) - step4: Rewrite in exponential form: \(2^{x-1}=2^{4}\) - step5: Set the exponents equal: \(x-1=4\) - step6: Move the constant to the right side: \(x=4+1\) - step7: Add the numbers: \(x=5\) Solve the equation \( 3^{2 x}-30(3^{x-2})+1=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(3^{2x}-30\times 3^{x-2}+1=0\) - step1: Factor the expression: \(\frac{1}{3}\left(3^{x}-3\right)\left(3^{x+1}-1\right)=0\) - step2: Calculate: \(\left(3^{x}-3\right)\left(3^{x+1}-1\right)=0\) - step3: Separate into possible cases: \(\begin{align}&3^{x}-3=0\\&3^{x+1}-1=0\end{align}\) - step4: Solve the equation: \(\begin{align}&x=1\\&x=-1\end{align}\) - step5: Rewrite: \(x_{1}=-1,x_{2}=1\) Solve the equation \( 2^{2 x}-5(2^{x})+2=0 \). Solve the equation by following steps: - step0: Solve for \(x\): \(2^{2x}-5\times 2^{x}+2=0\) - step1: Use substitution: \(t^{2}-5t+2=0\) - step2: Solve using the quadratic formula: \(t=\frac{5\pm \sqrt{\left(-5\right)^{2}-4\times 2}}{2}\) - step3: Simplify the expression: \(t=\frac{5\pm \sqrt{17}}{2}\) - step4: Separate into possible cases: \(\begin{align}&t=\frac{5+\sqrt{17}}{2}\\&t=\frac{5-\sqrt{17}}{2}\end{align}\) - step5: Substitute back: \(\begin{align}&2^{x}=\frac{5+\sqrt{17}}{2}\\&2^{x}=\frac{5-\sqrt{17}}{2}\end{align}\) - step6: Solve the equation for \(x:\) \(\begin{align}&x=\log_{2}{\left(5+\sqrt{17}\right)}-1\\&x=\log_{2}{\left(5-\sqrt{17}\right)}-1\end{align}\) - step7: Rewrite: \(x_{1}=\log_{2}{\left(5-\sqrt{17}\right)}-1,x_{2}=\log_{2}{\left(5+\sqrt{17}\right)}-1\) Here are the truth sets for each of the given equations: a) For the equation \( 2^{2 x+1}-5(2^{x})+2=0 \): - The solutions are \( x_{1} = -1 \) and \( x_{2} = 1 \). - **Truth set**: \( \{-1, 1\} \) b) For the equation \( 3^{2 x+3}-4(3^{x+1})+1=0 \): - The solutions are \( x_{1} = -2 \) and \( x_{2} = -1 \). - **Truth set**: \( \{-2, -1\} \) c) For the equation \( 3^{2 x}-30(3^{x-2})+1=0 \): - The solutions are \( x_{1} = -1 \) and \( x_{2} = 1 \). - **Truth set**: \( \{-1, 1\} \) d) For the equation \( 2^{2 x}+3(2^{x})-4=0 \): - The solution is \( x = 0 \). - **Truth set**: \( \{0\} \) e) For the equation \( 2^{2 x}-5(2^{x})+2=0 \): - The solutions are \( x_{1} = \log_{2}{(5-\sqrt{17})}-1 \) and \( x_{2} = \log_{2}{(5+\sqrt{17})}-1 \). - **Truth set**: \( \left\{ \log_{2}{(5-\sqrt{17})}-1, \log_{2}{(5+\sqrt{17})}-1 \right\} \) f) For the equation \( 3^{2 x+1}-26(3^{x})-9=0 \): - The solution is \( x = 2 \). - **Truth set**: \( \{2\} \) g) For the equation \( 4^{x}-6(2^{x})-16=0 \): - The solution is \( x = 3 \). - **Truth set**: \( \{3\} \) h) For the equation \( 2^{x}+2^{(x-1)}=48 \): - The solution is \( x = 5 \). - **Truth set**: \( \{5\} \) In summary, the truth sets for the equations are: - a) \( \{-1, 1\} \) - b) \( \{-2, -1\} \) - c) \( \{-1, 1\} \) - d) \( \{0\} \) - e) \( \left\{ \log_{2}{(5-\sqrt{17})}-1, \log_{2}{(5+\sqrt{17})}-1 \right\} \) - f) \( \{2\} \) - g) \( \{3\} \) - h) \( \{5\} \)