Discover new perspectives and gain insights with IDNLearn.com. Ask your questions and receive comprehensive, trustworthy responses from our dedicated team of experts.
Sagot :
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]
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]
Thank you for contributing to our discussion. Don't forget to check back for new answers. Keep asking, answering, and sharing useful information. Thank you for visiting IDNLearn.com. We’re here to provide dependable answers, so visit us again soon.
