If you encounter \(\sqrt{-5}\) while solving a quadratic equation, it means the equation has complex (or imaginary) solutions because the square root of a negative number is not real.

### Handling \(\sqrt{-5}\):
1. **Recognize the imaginary unit**: The square root of \(-1\) is represented by the imaginary unit \(i\), so \(\sqrt{-5}\) can be written as \(\sqrt{5} \cdot i\).
   \[
   \sqrt{-5} = \sqrt{5} \cdot i
   \]

2. **Substitute into the equation**: If you were solving an equation like \(x^2 + 6x + 10 = 0\) (which after completing the square would give you \(\sqrt{-5}\)), you would proceed as follows:

### Example:
Let's modify the original equation to create this scenario:

\[
x^2 + 6x + 10 = 0
\]

**Step 1**: Move the constant term to the right side:
\[
x^2 + 6x = -10
\]

**Step 2**: Complete the square:
\[
x^2 + 6x + 9 = -10 + 9
\]
\[
(x + 3)^2 = -1
\]

**Step 3**: Take the square root of both sides:
\[
x + 3 = \pm \sqrt{-1}
\]

**Step 4**: Substitute \(\sqrt{-1} = i\):
\[
x + 3 = \pm i
\]

**Step 5**: Isolate \(x\):
\[
x = -3 \pm i
\]

### Final Answer:
The solutions are \(x = -3 + i\) and \(x = -3 - i\). These are complex numbers.

### Summary:
When you encounter a negative number under the square root while solving a quadratic equation, the solutions will involve imaginary numbers, expressed in terms of \(i\).

Would you like further clarification? Here are some related questions:

1. What is the general form of a complex number?
2. How do you solve the quadratic equation \(x^2 + 4x + 5 = 0\) for complex solutions?
3. Can you explain how the discriminant determines whether the roots are real or complex?
4. What are the geometric interpretations of complex numbers on the complex plane?
5. How would you multiply and divide complex numbers like \(3 + 2i\) and \(1 - i\)?

**Tip:** When dealing with quadratic equations, the sign of the discriminant \(b^2 - 4ac\) determines the nature of the roots: positive for real and distinct, zero for real and equal, and negative for complex roots.

Explore how to handle √-5 in quadratic equations, understanding complex number solutions using the imaginary unit. Learn step-by-step with examples and detailed explanations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation