Solving the Equation 5^x + 5^x = 50Algebra problems from math competitions often provide excellent opportunities to develop and showcase problem-solving skills. Today, we’ll delve into solving the equation 5^x + 5^x = 50. This problem, while seemingly simple, requires a bit of algebraic insight to solve efficiently.
Algebra problems from math competitions often provide excellent opportunities to develop and showcase problem-solving skills. Today, we’ll delve into solving the equation 5^x + 5^x = 50. This problem, while seemingly simple, requires a bit of algebraic insight to solve efficiently.
Step-by-Step Solution
1. Understand the Problem:
The given equation is:
5^x + 5^x = 50
At first glance, it might seem complex, but the equation has a neat structure that simplifies our work.2. Combine Like Terms:
Notice that 5^x + 5^x can be thought of as adding the same term to itself:
5^x + 5^x = 2 * 5^x
This simplifies our equation to:
2*5^x = 503. Isolate the Exponential Term:
To solve for 5^x, we first isolate it by dividing both sides of the equation b2:
5^x = 50/2
5^x = 25
4. Solve for x:
Now, we recognize that 25 can be written as a power of 5:
25 = 5²
Therefore, we can set the exponents equal to each other since the bases are the same:
5^x = 5²
Thus:
x = 2
1. Understand the Problem:
The given equation is:
5^x + 5^x = 50
At first glance, it might seem complex, but the equation has a neat structure that simplifies our work.
2. Combine Like Terms:
Notice that 5^x + 5^x can be thought of as adding the same term to itself:
5^x + 5^x = 2 * 5^x
This simplifies our equation to:
2*5^x = 50
3. Isolate the Exponential Term:
To solve for 5^x, we first isolate it by dividing both sides of the equation b2:
5^x = 50/2
5^x = 25
4. Solve for x:
Now, we recognize that 25 can be written as a power of 5:
25 = 5²
Therefore, we can set the exponents equal to each other since the bases are the same:
5^x = 5²
Thus:
x = 2
Verification
It’s always good practice to verify the solution by substituting it back into the original equation. Let’s check x=2:
5^x + 5^x = 5² + 5² = 25 + 25 = 50
The left side of the equation matches the right side, confirming that our solution is correct.
It’s always good practice to verify the solution by substituting it back into the original equation. Let’s check x=2:
5^x + 5^x = 5² + 5² = 25 + 25 = 50
The left side of the equation matches the right side, confirming that our solution is correct.
Conclusion
The solution to the equation 5^x + 5^x = 50 is x = 2. This problem serves as an excellent example of how recognizing patterns and simplifying expressions can make solving algebraic equations straightforward. It’s also a reminder that breaking down problems into smaller, more manageable parts often leads to the solution. Happy problem-solving!
The solution to the equation 5^x + 5^x = 50 is x = 2. This problem serves as an excellent example of how recognizing patterns and simplifying expressions can make solving algebraic equations straightforward. It’s also a reminder that breaking down problems into smaller, more manageable parts often leads to the solution. Happy problem-solving!
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