why does the equation x^2=-9 has no real solution? This I can't figure out.

why does the equation x^2 = 9 has two solutions? Is it because it satisfied by both 3 and -3 so it has two solutions.

Whether x = 3 or -3, its square (3*3) or (-3*-3) will always be positive, not negative.

x^2 = 9, so x = 3 or -3

Thank you so very much.

To understand why the equation x^2 = -9 has no real solutions, let's break it down step by step.

1. Start with the equation x^2 = -9.
2. Take the square root of both sides to isolate x: sqrt(x^2) = sqrt(-9).
3. The square root of a positive number gives both a positive and a negative solution. However, the square root of a negative number is not a real number, but rather an imaginary number.
4. To represent the square root of -9, we can write it as √(-9) = √9 * √(-1) = 3i. Here, the symbol 'i' represents the imaginary unit (√(-1)).
5. So the solutions to the equation x^2 = -9 are x = 3i and x = -3i. These are complex solutions, involving imaginary numbers.

Now, let's consider the equation x^2 = 9.

1. Start with the equation x^2 = 9.
2. Take the square root of both sides: sqrt(x^2) = sqrt(9).
3. The square root of a positive number gives both a positive and a negative solution.
4. So we have ±√9 = ±3. Here, the plus-minus symbol (±) indicates that both positive and negative solutions are valid.
5. Therefore, the solutions to the equation x^2 = 9 are x = 3 and x = -3.

In summary, the equation x^2 = -9 has no real solutions because it involves the square root of a negative number, resulting in complex (imaginary) solutions. On the other hand, the equation x^2 = 9 has two solutions because it satisfies both positive (x = 3) and negative (x = -3) values, resulting in two real solutions.