1.5 \( 2 \sqrt{2-7 x}=\sqrt{-36 x} \) \( \sqrt{2-x}-x=-2 \) .7 \( 26-5^{2 x}=\left(5^{x}-6\right)^{2} \) Show that the equation \( 2^{2 x+1}+7.2^{x}-4=0 \) has only ONE solution. SIMPLIFICATION
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Let's embark on a little adventure with equations! The first equation \( 2 \sqrt{2-7 x}=\sqrt{-36 x} \) tells us to solve for \( x \). By squaring both sides (and keeping in mind to check for extraneous solutions later), you can simplify things. Just remember: inequalities under square roots can lead to interesting domains! Now, regarding the equation \( 2^{2 x+1}+7.2^{x}-4=0 \), you can help yourself by making a substitution such as \( y = 2^x \). This transforms the problem into a quadratic equation, which is much easier to tackle! Analyze the quadratic's discriminant to show it has just one solution by ensuring it is zero – and voilà, you've proven uniqueness!