To solve the quadratic equation $x^2 - 4x + 12 = 0$ using the quadratic formula, we follow these steps:

Quadratic Formula:

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

Given Equation:

$x^2 - 4x + 12 = 0$

Identify the coefficients:

• $a = 1$
• $b = -4$
• $c = 12$

Step 1: Calculate the discriminant ($\Delta$):

$\Delta = b^2 - 4ac = (-4)^2 - 4(1)(12) = 16 - 48 = -32$

Step 2: Substitute the values into the quadratic formula:

$x = \frac{-(-4) \pm \sqrt{-32}}{2(1)} = \frac{4 \pm \sqrt{-32}}{2}$

Step 3: Simplify the expression:

$x = \frac{4 \pm \sqrt{32} \cdot i}{2}$

Step 4: Further simplify:

$x = \frac{4 \pm 4\sqrt{2} \cdot i}{2}$ $x = 2 \pm 2\sqrt{2} \cdot i$

Final Answer:

$x = 2 + 2\sqrt{2} \cdot i \quad \text{or} \quad x = 2 - 2\sqrt{2} \cdot i$

These are the two complex roots of the equation.

Would you like more details or have any questions?

Here are five related questions:

1. What is the significance of the discriminant in the quadratic formula?
2. How do complex roots differ from real roots?
3. Can the quadratic formula be used for any quadratic equation?
4. What are some practical applications of complex numbers?
5. How can we graph a quadratic function with complex roots?

Tip: If the discriminant is negative, the roots of the quadratic equation will be complex, involving the imaginary unit $i$.