Math Problem Statement
Solution
The equation provided is:
This can be solved by using a substitution to simplify the equation. Let's follow these steps:
Step 1: Substitution
Let . This gives us the following substitution: So the original equation becomes:
Step 2: Solve the quadratic equation
Now we solve the quadratic equation: We can factor this as:
Step 3: Solve for
From the factored equation, we have two possible solutions:
Step 4: Solve for
Recall that , so now we solve for in each case:
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If , then . Since , we have:
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If , then . This gives:
Final Solutions:
The solutions for are:
Let me know if you need more details or clarifications! Here are some related questions:
- How do we solve other types of exponential equations?
- What are common substitution methods for exponential equations?
- How can we check if the solutions to an equation are correct?
- How can we use logarithms to solve exponential equations?
- What are the properties of exponents that simplify solving exponential equations?
Tip: When faced with exponential equations, try using substitution to reduce the complexity of the problem before solving.
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Math Problem Analysis
Mathematical Concepts
Algebra
Exponential Equations
Quadratic Equations
Formulas
Quadratic equation formula ax^2 + bx + c = 0
Theorems
Quadratic formula
Suitable Grade Level
Grades 9-11