To solve the equation [tex]\(6^x + 6^{-x} = 6 \frac{1}{6}\)[/tex], let's follow these steps:
1. Write the given equation:
[tex]\[
6^x + 6^{-x} = 6 \frac{1}{6}
\][/tex]
2. Convert the mixed number [tex]\(6 \frac{1}{6}\)[/tex] into an improper fraction:
[tex]\[
6 \frac{1}{6} = \frac{6 \cdot 6 + 1}{6} = \frac{37}{6}
\][/tex]
So, the equation becomes:
[tex]\[
6^x + 6^{-x} = \frac{37}{6}
\][/tex]
3. Let [tex]\(y = 6^x\)[/tex]. This implies that [tex]\(6^{-x} = \frac{1}{y}\)[/tex]. Substitute these into the equation:
[tex]\[
y + \frac{1}{y} = \frac{37}{6}
\][/tex]
4. Multiply both sides by [tex]\(y\)[/tex] to eliminate the fraction:
[tex]\[
y^2 + 1 = \frac{37}{6} y
\][/tex]
5. Rearrange the equation to form a standard quadratic equation:
[tex]\[
6y^2 + 6 = 37y
\][/tex]
[tex]\[
6y^2 - 37y + 6 = 0
\][/tex]
6. Use the quadratic formula to solve for [tex]\( y \)[/tex]:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\(a = 6\)[/tex], [tex]\(b = -37\)[/tex], and [tex]\(c = 6\)[/tex].
7. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac = (-37)^2 - 4 \cdot 6 \cdot 6 = 1369 - 144 = 1225
\][/tex]
8. Find the square root of the discriminant:
[tex]\[
\sqrt{1225} = 35
\][/tex]
9. Substitute these values into the quadratic formula:
[tex]\[
y = \frac{37 \pm 35}{12}
\][/tex]
10. Solve for the two possible values of [tex]\( y \)[/tex]:
[tex]\[
y_1 = \frac{37 + 35}{12} = \frac{72}{12} = 6
\][/tex]
[tex]\[
y_2 = \frac{37 - 35}{12} = \frac{2}{12} = \frac{1}{6}
\][/tex]
11. Since [tex]\(y = 6^x\)[/tex] and [tex]\(y\)[/tex] must be positive, we take [tex]\(y = 6\)[/tex] (as [tex]\(6^x = \frac{1}{6}\)[/tex] would imply a negative [tex]\(x\)[/tex] and is still valid mathematically).
12. Solve for [tex]\( x \)[/tex] using [tex]\(6^x = 6\)[/tex]:
[tex]\[
x = 1
\][/tex]
Thus, the solutions to the equation [tex]\(6^x + 6^{-x} = 6 \frac{1}{6}\)[/tex] are:
- [tex]\(y_1 = 6\)[/tex], [tex]\(y_2 = \frac{1}{6}\)[/tex]
- The value of [tex]\( y \)[/tex] taken is 6
- [tex]\(x = 1\)[/tex]
