What are the roots of the equation x, squared, plus, 4, x, plus, 29, equals, 0*x*2+4*x*+29=0 in simplest a, plus, b, i*a*+*bi* form?

To find the roots of the quadratic equation \(x^2 + 4x + 29 = 0\) in the form \(a + bi\), we will use the quadratic formula:

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

Given the equation \(x^2 + 4x + 29 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 4\)
- \(c = 29\)

### Step 1: Calculate the discriminant
The discriminant \(\Delta\) is:

\[
\Delta = b^2 - 4ac
\]

Substituting the values:

\[
\Delta = (4)^2 - 4(1)(29) = 16 - 116 = -100
\]

### Step 2: Solve for the roots
Since the discriminant is negative, the roots are complex. Applying the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{-100}}{2(1)}
\]

The square root of \(-100\) is \(10i\), so:

\[
x = \frac{-4 \pm 10i}{2}
\]

### Step 3: Simplify the expression
Divide both terms by 2:

\[
x = -2 \pm 5i
\]

### Final Answer:
The roots of the equation are \(x = -2 + 5i\) and \(x = -2 - 5i\), in the form \(a + bi\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you determine whether the roots of a quadratic equation will be real or complex?
2. What is the importance of the imaginary unit \(i\) in complex numbers?
3. Can the quadratic formula be used for any type of quadratic equation?
4. How would you represent the roots on a complex plane?
5. What is the relationship between the sum and product of the roots and the coefficients of the quadratic equation?

**Tip:** When dealing with complex roots, remember that they always come in conjugate pairs (e.g., \(a + bi\) and \(a - bi\)).

Learn how to solve the quadratic equation x^2 + 4x + 29 = 0 in the simplest a + bi form. Understand the concept of complex roots, use of the discriminant, and application of the quadratic formula. Suitable for advanced high school students and college-level learners.

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