To find the roots of the quadratic equation $4x^2 - 12x + 13 = 0$, we will use the quadratic formula:

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

where $a = 4$, $b = -12$, and $c = 13$.

Step 1: Calculate the discriminant

The discriminant is given by $b^2 - 4ac$:

$(-12)^2 - 4(4)(13) = 144 - 208 = -64$

Since the discriminant is negative, the roots will be complex.

Step 2: Apply the quadratic formula

Substitute the values into the formula:

$x = \frac{-(-12) \pm \sqrt{-64}}{2 \times 4} = \frac{12 \pm \sqrt{-64}}{8}$

Step 3: Simplify the square root of the negative number

$\sqrt{-64}$ can be written as $8i$, where $i$ is the imaginary unit.

So, the equation becomes:

$x = \frac{12 \pm 8i}{8} = \frac{12}{8} \pm \frac{8i}{8} = \frac{3}{2} \pm i$

Final Answer

The roots in $a + bi$ form are:

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

Let me know if you need further details or explanations.

Follow-Up Questions:

1. How does the discriminant determine the nature of the roots?
2. What happens if the discriminant is zero in a quadratic equation?
3. How do you find real roots using the quadratic formula?
4. Can the quadratic formula solve equations with coefficients that are not integers?
5. How do complex roots relate to the graph of a quadratic equation?

Tip:

For quadratic equations, always check the discriminant ($b^2 - 4ac$) to determine if the roots are real or complex before proceeding.