To solve the quadratic equation:

$x^2 - 4x + 7 = 0$

we can use the quadratic formula:

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

For this equation, the coefficients are:

• $a = 1$
• $b = -4$
• $c = 7$

Step 1: Substitute the values into the formula

$x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 1 \cdot 7}}{2 \cdot 1}$

Simplifying, we get:

$x = \frac{4 \pm \sqrt{16 - 28}}{2}$

Step 2: Simplify inside the square root

$x = \frac{4 \pm \sqrt{-12}}{2}$

Since the discriminant ($-12$) is negative, we will have complex roots. We rewrite $\sqrt{-12}$ as $2i\sqrt{3}$, where $i$ is the imaginary unit.

$x = \frac{4 \pm 2i\sqrt{3}}{2}$

Step 3: Simplify further

Dividing both terms in the numerator by 2:

$x = 2 \pm i\sqrt{3}$

Final Answer

The two complex roots in $a + bi$ form are:

$x = 2 + i\sqrt{3} \quad \text{and} \quad x = 2 - i\sqrt{3}$

Would you like further details on any steps? Here are some related questions:

1. How do we know when to expect complex roots in a quadratic equation?
2. What is the role of the discriminant in determining the nature of roots?
3. How is the imaginary unit $i$ defined in complex numbers?
4. How can complex conjugates appear as roots in quadratic equations?
5. Why is the quadratic formula useful for equations with complex solutions?

Tip: Always check the discriminant ($b^2 - 4ac$) to quickly determine if the roots will be real or complex.