Explanation:
To find the roots of the equation 4x^2 - 8x + 13 = 0, we can use the quadratic formula. The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the roots can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
Let's apply this formula to our equation:
a = 4
b = -8
c = 13
Plugging these values into the quadratic formula, we get:
x = (-(-8) ± √((-8)^2 - 4(4)(13))) / (2(4))
Simplifying further:
x = (8 ± √(64 - 208)) / 8
x = (8 ± √(-144)) / 8
Since we have a negative value under the square root, the roots of this equation will involve complex numbers. Let's continue simplifying:
x = (8 ± √(144) * √(-1)) / 8
x = (8 ± 12i) / 8
Now, dividing both numerator and denominator by 4:
x = (2 ± 3i) / 2
Finally, we can simplify further by factoring out 2 from the numerator:
x = 2(1 ± 3i) / 2
Canceling out the common factor of 2:
x = 1 ± 3i
Therefore, the roots of the equation 4x^2 - 8x + 13 = 0 in simplest a + bi form are 1 + 3i and 1 - 3i.