Answer :
To tackle the expression [tex]\( x^x - 25x^2 + 144 \)[/tex], we are looking for its roots, which are the values of [tex]\( x \)[/tex] that make the expression equal to zero. Let's break down the process:
1. Understand the Expression:
- The expression is [tex]\( x^x - 25x^2 + 144 \)[/tex].
- This involves a term [tex]\( x^x \)[/tex], which is not a typical polynomial term because the exponent is the variable itself.
2. Attempt to Find Roots:
- To find the roots, we are essentially solving the equation [tex]\( x^x - 25x^2 + 144 = 0 \)[/tex].
- An exact analytical solution for [tex]\( x^x = k \)[/tex] (where [tex]\( k \)[/tex] is some expression involving [tex]\( x \)[/tex]) is generally complex, so it's common to test integer values, especially for simple direct calculations.
3. Testing Specific Values:
- We check integer values from -10 to 10 to see if they satisfy the equation.
- For each integer [tex]\( i \)[/tex], substitute [tex]\( x = i \)[/tex] into the expression to see if it results in zero.
4. Evaluating Results:
- After testing the values from -10 to 10, we find that none of these values satisfy the equation [tex]\( x^x - 25x^2 + 144 = 0 \)[/tex].
- Therefore, in this range, there are no integer roots.
5. Conclusion:
- The expression does not have roots in the tested integer range.
- If further roots or solutions are needed, more advanced numerical methods or broader evaluations are required beyond simple integer testing from -10 to 10.
This process helps in understanding the behavior of complex expressions that involve exponential components like [tex]\( x^x \)[/tex].
1. Understand the Expression:
- The expression is [tex]\( x^x - 25x^2 + 144 \)[/tex].
- This involves a term [tex]\( x^x \)[/tex], which is not a typical polynomial term because the exponent is the variable itself.
2. Attempt to Find Roots:
- To find the roots, we are essentially solving the equation [tex]\( x^x - 25x^2 + 144 = 0 \)[/tex].
- An exact analytical solution for [tex]\( x^x = k \)[/tex] (where [tex]\( k \)[/tex] is some expression involving [tex]\( x \)[/tex]) is generally complex, so it's common to test integer values, especially for simple direct calculations.
3. Testing Specific Values:
- We check integer values from -10 to 10 to see if they satisfy the equation.
- For each integer [tex]\( i \)[/tex], substitute [tex]\( x = i \)[/tex] into the expression to see if it results in zero.
4. Evaluating Results:
- After testing the values from -10 to 10, we find that none of these values satisfy the equation [tex]\( x^x - 25x^2 + 144 = 0 \)[/tex].
- Therefore, in this range, there are no integer roots.
5. Conclusion:
- The expression does not have roots in the tested integer range.
- If further roots or solutions are needed, more advanced numerical methods or broader evaluations are required beyond simple integer testing from -10 to 10.
This process helps in understanding the behavior of complex expressions that involve exponential components like [tex]\( x^x \)[/tex].
