To solve the equation [tex]\(\sqrt{x + 10} = -5\)[/tex], we need to understand the properties of the square root function.
1. Understanding the Square Root Function:
- The square root function, [tex]\(\sqrt{y}\)[/tex], always produces a non-negative result for any real number [tex]\(y\)[/tex].
- This means [tex]\(\sqrt{y} \geq 0\)[/tex] for all [tex]\(y \geq 0\)[/tex].
2. Given Equation:
- The equation given is [tex]\(\sqrt{x + 10} = -5\)[/tex].
- Here, we need to determine whether this equation can make sense in the realm of real numbers.
3. Analyzing the Right Side of the Equation:
- The right side of the equation is [tex]\(-5\)[/tex], which is a negative number.
4. Comparing with Properties of Square Root:
- Since the square root function always produces a non-negative result ([tex]\(\sqrt{y} \geq 0\)[/tex]), it can never equal a negative number.
- Specifically, [tex]\(\sqrt{x + 10} \geq 0\)[/tex] for any real value of [tex]\(x\)[/tex].
5. Conclusion:
- As the left side of the equation represents a non-negative number and the right side is a negative number, there is a contradiction.
- It implies that there is no real number [tex]\(x\)[/tex] that can satisfy the equation [tex]\(\sqrt{x + 10} = -5\)[/tex].
Therefore, the final conclusion is: There is no solution.
