solve quadratic equation with complex roots
Check the final answer first, then review the worked steps.
Problem
solve quadratic equation with complex roots
Answer
\(5i, -5i\)
Step-by-step solution
- Identify the equation: We are given the quadratic equation $x^2 = -25$ and asked to solve for $x$ in the set of complex numbers.
- Apply the square root property: To solve for $x$, we take the square root of both sides of the equation. Remember that when taking the square root of both sides, we must include both the positive and negative roots: $x = \pm \sqrt{-25}$.
- Use the definition of the imaginary unit: In the set of complex numbers, the imaginary unit $i$ is defined as $i = \sqrt{-1}$. Therefore, we can rewrite the expression under the square root as: $x = \pm \sqrt{25 \cdot (-1)}$.
- Simplify the expression: Using the property of radicals $\sqrt{ab} = \sqrt{a} \cdot \sqrt{b}$, we get: $x = \pm \sqrt{25} \cdot \sqrt{-1}$. Since $\sqrt{25} = 5$ and $\sqrt{-1} = i$, the solutions are $x = 5i$ and $x = -5i$.