When solving quadratic equations using the square root property, encountering a negative number under the square root indicates the presence of imaginary or complex solutions. This occurs because the square root of a negative number is not defined within the real numbers but can be expressed using the imaginary unit i, where i is defined as the square root of -1, or \(i = \sqrt{-1}\).
For example, consider the quadratic equation \$2x^2 + 32 = 0\(. To solve for x, first isolate the squared term by subtracting 32 from both sides:
\[2x^2 = -32\]Next, divide both sides by 2 to simplify:
\[x^2 = -16\]Applying the square root property, take the square root of both sides:
\[x = \pm \sqrt{-16}\]Since the radicand (the number under the square root) is negative, rewrite the square root using the product property:
\[\sqrt{-16} = \sqrt{16} \times \sqrt{-1}\]Recognizing that \)\sqrt{16} = 4\( and \)\sqrt{-1} = i\(, the expression simplifies to:
\[x = \pm 4i\]Thus, the solutions are complex numbers: \)x = 4i\( and \)x = -4i$. This method demonstrates how the square root property extends to complex solutions by incorporating the imaginary unit i. Understanding this process is essential for solving quadratic equations that do not have real roots, expanding the scope of solutions to include the complex number system.