What are the roots of the equation x, squared, minus, 16, x, plus, 68, equals, 0x 
2
 −16x+68=0 in simplest a, plus, b, ia+bi form?

The given equation is:

\[
x^2 - 16x + 68 = 0
\]

To solve for the roots in \( a + bi \) form (where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit), we will use the **quadratic formula**:

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

Here, \( a = 1 \), \( b = -16 \), and \( c = 68 \).

Let's apply the formula step-by-step:

1. **Calculate the discriminant**:
   \[
   \Delta = b^2 - 4ac = (-16)^2 - 4(1)(68) = 256 - 272 = -16
   \]

2. **Since the discriminant is negative**, the solutions will involve complex numbers.

3. **Solve using the quadratic formula**:
   \[
   x = \frac{-(-16) \pm \sqrt{-16}}{2(1)} = \frac{16 \pm \sqrt{-16}}{2}
   \]

4. **Simplify the square root of the negative discriminant**:
   \[
   \sqrt{-16} = 4i
   \]

5. **Substitute back into the quadratic formula**:
   \[
   x = \frac{16 \pm 4i}{2} = 8 \pm 2i
   \]

So, the two roots of the equation are:

\[
x_1 = 8 + 2i \quad \text{and} \quad x_2 = 8 - 2i
\]

### Final Answer: The roots are \( 8 + 2i \) and \( 8 - 2i \).

Would you like more details on any of these steps?

Here are 5 related questions for further exploration:
1. How is the discriminant related to the nature of roots?
2. What is the significance of the quadratic formula in solving any quadratic equation?
3. How do imaginary numbers arise when the discriminant is negative?
4. Can complex numbers have real components only?
5. What is the geometric interpretation of complex roots?

**Tip**: When the discriminant is negative, the roots will always be complex (involving imaginary numbers).

What are the roots of the equation x^2 - 16x + 68 = 0 in simplest a + bi form?

Discover how to solve the quadratic equation x^2 - 16x + 68 using the quadratic formula. This problem involves complex numbers and explores key concepts like the discriminant. Ideal for students in Grades 9-12, this step-by-step solution provides insight into solving quadratic equations with complex roots.

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