To solve the equation [tex]\(5^x + \frac{1}{5^x} = 5 + \frac{1}{5}\)[/tex], we will go through a structured and detailed solution step by step.
1. Rewrite the equation:
Start with the equation:
[tex]\[
5^x + \frac{1}{5^x} = 5 + \frac{1}{5}
\][/tex]
Notice that [tex]\(5 + \frac{1}{5}\)[/tex] can be simplified:
[tex]\[
5 + \frac{1}{5} = \frac{25}{5} + \frac{1}{5} = \frac{26}{5}
\][/tex]
So the equation becomes:
[tex]\[
5^x + \frac{1}{5^x} = \frac{26}{5}
\][/tex]
2. Introduce a substitution:
Let [tex]\(y = 5^x\)[/tex]. This helps to simplify the equation:
[tex]\[
y + \frac{1}{y} = \frac{26}{5}
\][/tex]
3. Eliminate the fraction:
Multiply both sides of the equation by [tex]\(y\)[/tex] to clear the fraction:
[tex]\[
y^2 + 1 = \frac{26}{5} y
\][/tex]
Multiply the whole equation by 5 to clear the denominator:
[tex]\[
5y^2 + 5 = 26y
\][/tex]
4. Rearrange into a standard quadratic equation form:
[tex]\[
5y^2 - 26y + 5 = 0
\][/tex]
5. Solve the quadratic equation:
The quadratic equation in standard form is [tex]\(ay^2 + by + c = 0\)[/tex], where [tex]\(a = 5\)[/tex], [tex]\(b = -26\)[/tex], and [tex]\(c = 5\)[/tex]. Use the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Calculate the discriminant first:
[tex]\[
\Delta = b^2 - 4ac = (-26)^2 - 4 \cdot 5 \cdot 5 = 676 - 100 = 576
\][/tex]
The square root of the discriminant:
[tex]\[
\sqrt{576} = 24
\][/tex]
Now, solve for [tex]\(y\)[/tex]:
[tex]\[
y = \frac{26 \pm 24}{10}
\][/tex]
This gives us two solutions:
[tex]\[
y_1 = \frac{26 + 24}{10} = \frac{50}{10} = 5
\][/tex]
[tex]\[
y_2 = \frac{26 - 24}{10} = \frac{2}{10} = 0.2
\][/tex]
6. Back-substitute to find [tex]\(x\)[/tex]:
We earlier set [tex]\(y = 5^x\)[/tex]. Now solve each [tex]\(y\)[/tex] value for [tex]\(x\)[/tex]:
For [tex]\(y_1 = 5\)[/tex]:
[tex]\[
5 = 5^x \implies x = 1
\][/tex]
For [tex]\(y_2 = 0.2\)[/tex]:
Note that [tex]\(0.2 = \frac{1}{5}\)[/tex]. Thus:
[tex]\[
\frac{1}{5} = 5^x \implies 5^{-1} = 5^x \implies x = -1
\][/tex]
Therefore, the solutions to the equation [tex]\(5^x + \frac{1}{5^x} = 5 + \frac{1}{5}\)[/tex] are:
[tex]\[
x = 1 \quad \text{and} \quad x = -1
\][/tex]
