Alright, let's start by writing down the original quadratic equation:
z^(2)-8z+13=0
To solve this equation, we will use the quadratic formula. The quadratic formula for solving an equation in the form of ax^2 + bx + c = 0 is as follows:
z = [ -b ± sqrt(b^2 - 4ac) ] / 2a
In our problem, a is the coefficient of z^2 which is 1, b is the coefficient of z which is -8, and c is the constant term which is 13.
Let's plug these values into the quadratic formula. But before we do that, let's calculate the discriminant, which is the term under the square root in the formula.
The discriminant is given by b^2 - 4ac. So, in our case, it becomes (-8)^2 - 4*1*13 = 64 - 52 = 12
Now, we use the quadratic formula to calculate the two possible values for z:
z1 = [ -(-8) + sqrt(12) ] / 2*1 = [ 8 + sqrt(12) ] / 2
= [ 8 + 3.46 ] / 2
≈ 5.73
z2 = [ -(-8) - sqrt(12) ] / 2*1 = [ 8 - sqrt(12) ] / 2
= [ 8 - 3.46 ] / 2
≈ 2.27
So, the solutions for the quadratic equation z^(2)-8z+13=0 using the quadratic formula are z1 ≈ 5.73 and z2 ≈ 2.27.