A student claims that the equation√-x=3 has no solution, since the square root of a negative number does not exist. Why is this argument wrong?
What if x itself is negative ?
e.g. √-(-9) = 3
The student's argument that the equation √-x = 3 has no solution because the square root of a negative number does not exist is incorrect. Although the square root of a negative number is not defined in the realm of real numbers, it does exist in the realm of complex numbers. Therefore, to properly address this argument, we need to consider the solution in the complex number system.
To find the solution to the equation, let's work through it step-by-step:
1. Start with the equation √-x = 3.
2. Square both sides of the equation to eliminate the square root: (√-x)^2 = (3)^2.
3. Simplify the left side of the equation: -x = 9.
4. Multiply both sides of the equation by -1 to isolate x: -1 * -x = -1 * 9.
5. Simplify: x = -9.
Therefore, the solution to the equation √-x = 3 is x = -9. Even though the original equation involves the square root of a negative number, we were able to find a solution in the form of a real number by taking the square root of the equation and moving through the steps. It's important to note that this solution exists within the realm of complex numbers.
In conclusion, while the square root of a negative number is not defined in the real number system, we can find solutions by considering complex numbers.