Math Problem Statement
Solution
To solve this problem, let's go through it step-by-step.
We are given the equation:
and we are told that and are the solutions to this equation. We need to find the value of .
Step 1: Simplify the Equation
The left-hand side of the equation can be simplified using properties of exponents. Let's express and in terms of base powers:
Thus, we can rewrite the equation as:
Using the power of a power rule , we get:
Now, we can combine the exponents:
Simplify the exponent:
Step 2: Equate Exponents
Since and are reciprocal values, we can set the exponent equal to (since ):
Add 1 to both sides:
Step 3: Solve the Quadratic Equation
Now we have a quadratic equation:
Using the quadratic formula , where , , and :
This gives us two solutions:
Step 4: Find
Now that we have and , let's find :
Then,
Answer
The answer is:
Do you have any questions or need further details on any step?
Related Questions
- How would the solution change if the base in the equation was different?
- Can we solve this equation using logarithms instead of simplifying the bases?
- What if we were asked to find instead of ?
- How does the discriminant in the quadratic formula help us determine the nature of the roots?
- What other methods can be used to solve quadratic equations?
Tip
When working with exponentials, try expressing terms with common bases to simplify the equation.
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Math Problem Analysis
Mathematical Concepts
Algebra
Quadratic Equations
Exponentials
Formulas
Quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Theorems
Exponent rules
Quadratic formula theorem
Suitable Grade Level
Grades 10-12
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