What are the roots of the equation x, squared, minus, 4, x, plus, 68, equals, 0x
2
−4x+68=0 in simplest a, plus, b, ia+bi form?

To find the roots of the quadratic equation \(x^2 - 4x + 68 = 0\), we'll use the quadratic formula:

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

In the equation \(x^2 - 4x + 68 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -4\)
- \(c = 68\)

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

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

Substitute the values of \(a\), \(b\), and \(c\):

\[
\Delta = (-4)^2 - 4(1)(68) = 16 - 272 = -256
\]

### Step 2: Solve for the roots
Since the discriminant is negative, the roots will be complex. Using the quadratic formula:

\[
x = \frac{-(-4) \pm \sqrt{-256}}{2(1)}
\]

Simplify this:

\[
x = \frac{4 \pm \sqrt{-256}}{2}
\]

\(\sqrt{-256}\) can be written as \(16i\), where \(i\) is the imaginary unit:

\[
x = \frac{4 \pm 16i}{2}
\]

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

\[
x = 2 \pm 8i
\]

### Final Answer:
The roots of the equation are \(x = 2 + 8i\) and \(x = 2 - 8i\), in the form \(a + bi\).

Would you like more details or have any questions?

Here are 5 related questions:
1. How do you interpret the discriminant in the context of complex roots?
2. What are the real and imaginary parts of the roots?
3. How would the equation change if the discriminant were positive?
4. Can you graphically represent complex roots on the complex plane?
5. How does the quadratic formula derive from completing the square?

**Tip:** For any quadratic equation, always check the discriminant first to determine the nature of the roots (real or complex).

Discover how to find the roots of the quadratic equation x^2 - 4x + 68 = 0 in the form a + bi. This detailed solution uses the quadratic formula and explores complex numbers. Suitable for advanced high school students.

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