To solve the equation [tex]\(16^{x+1} - 24 \times 4^x + 8 = 0\)[/tex], let's follow these steps:
1. Rewrite the equation using exponent properties:
[tex]\[16^{x+1} = (2^4)^{x+1} = 2^{4(x+1)} = 2^{4x + 4}\][/tex]
[tex]\[4^x = (2^2)^x = 2^{2x}\][/tex]
Substituting these into the equation gives:
[tex]\[2^{4x + 4} - 24 \times 2^{2x} + 8 = 0\][/tex]
2. Simplify exponent handling:
Let [tex]\(y = 2^{2x}\)[/tex]. Therefore:
[tex]\[2^{4x+4} = 2^4 \cdot 2^{4x} = 16 \cdot (2^{2x})^2 = 16y^2\][/tex]
Substituting [tex]\(y = 2^{2x}\)[/tex]:
[tex]\[16y^2 - 24y + 8 = 0\][/tex]
3. Solve the quadratic equation:
[tex]\[16y^2 - 24y + 8 = 0\][/tex]
This is a standard quadratic equation of the form [tex]\(ay^2 + by + c = 0\)[/tex]. Using the quadratic formula:
[tex]\[ y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 16\)[/tex], [tex]\(b = -24\)[/tex], and [tex]\(c = 8\)[/tex]. Plugging in these values:
[tex]\[ y = \frac{24 \pm \sqrt{(-24)^2 - 4 \cdot 16 \cdot 8}}{2 \cdot 16} \][/tex]
[tex]\[ y = \frac{24 \pm \sqrt{576 - 512}}{32} \][/tex]
[tex]\[ y = \frac{24 \pm \sqrt{64}}{32} \][/tex]
[tex]\[ y = \frac{24 \pm 8}{32} \][/tex]
This provides two solutions:
[tex]\[ y_1 = \frac{24 + 8}{32} = \frac{32}{32} = 1 \][/tex]
[tex]\[ y_2 = \frac{24 - 8}{32} = \frac{16}{32} = \frac{1}{2} \][/tex]
4. Back-substitute [tex]\( y = 2^{2x} \)[/tex]:
For [tex]\( y_1 = 1 \)[/tex]:
[tex]\[ 2^{2x} = 1 \][/tex]
[tex]\[ 2x = 0 \][/tex]
[tex]\[ x = 0 \][/tex]
For [tex]\( y_2 = \frac{1}{2} \)[/tex]:
[tex]\[ 2^{2x} = \frac{1}{2} \][/tex]
[tex]\[ 2x = \log_2 \left( \frac{1}{2} \right) \][/tex]
[tex]\[ 2x = -1 \][/tex]
[tex]\[ x = -\frac{1}{2} \][/tex]
5. Include complex solutions in real-life scenarios, exponentiation equations can yield non-real (complex) solutions. Additional steps using complex analysis would reveal the additional roots:
[tex]\[ x = \frac{i\pi}{\log(2)} \][/tex]
[tex]\[ x = -\frac{1}{2} + \frac{i\pi}{\log(2)} \][/tex]
Combining all these results, the complete solution set for the equation [tex]\(16^{x+1} - 24 \times 4^x + 8 = 0\)[/tex] is:
[tex]\[ \boxed{\left\{-\frac{1}{2}, 0, \frac{i\pi}{\log(2)}, -\frac{1}{2} + \frac{i\pi}{\log(2)}\right\}} \][/tex]
