What are the roots of the equation x, squared, plus, 14, x, plus, 74, equals, 0*x*2+14*x*+74=0 in simplest a, plus, b, i*a*+*bi* form?

To find the roots of the quadratic equation \( x^2 + 14x + 74 = 0 \), we can use the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

In this equation:
- \( a = 1 \)
- \( b = 14 \)
- \( c = 74 \)

### Step 1: Calculate the Discriminant
The discriminant is \( b^2 - 4ac \):

\[
b^2 - 4ac = 14^2 - 4 \cdot 1 \cdot 74 = 196 - 296 = -100
\]

Since the discriminant is negative, the roots will be complex.

### Step 2: Substitute into the Quadratic Formula
Substitute the values into the formula:

\[
x = \frac{-14 \pm \sqrt{-100}}{2 \cdot 1} = \frac{-14 \pm \sqrt{100} \cdot i}{2} = \frac{-14 \pm 10i}{2}
\]

### Step 3: Simplify
Divide each term by 2:

\[
x = -7 \pm 5i
\]

### Final Answer
The roots of the equation are:
\[
x = -7 + 5i \quad \text{and} \quad x = -7 - 5i
\]

Would you like additional details or explanations on any steps?

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Here are some questions to explore further:
1. How would you find the roots if the discriminant were positive?
2. What does a negative discriminant indicate about the graph of the quadratic function?
3. How can complex roots be used in physics or engineering problems?
4. What are the implications of complex roots in signal processing?
5. How does the quadratic formula apply to higher-degree polynomials?

**Tip:** When working with complex roots, remember that they always come in conjugate pairs if the coefficients of the polynomial are real.

What are the roots of the equation x^2 + 14x + 74 = 0 in simplest a + bi form?

Explore how to solve the quadratic equation x^2 + 14x + 74 = 0 and find its roots in the form a + bi. This solution provides a detailed step-by-step approach, using the quadratic formula and addressing complex roots. Suitable for high school algebra students.

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