To solve the quadratic equation [tex]\(5x^2 - 8x + 5 = 0\)[/tex] using the quadratic formula, let’s go through the steps in a detailed manner.
The quadratic formula is given by:
[tex]\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, the coefficients are [tex]\(a = 5\)[/tex], [tex]\(b = -8\)[/tex], and [tex]\(c = 5\)[/tex].
First, we calculate the discriminant, [tex]\(\Delta\)[/tex]:
[tex]\[
\Delta = b^2 - 4ac = (-8)^2 - 4 \cdot 5 \cdot 5 = 64 - 100 = -36
\][/tex]
Since the discriminant is negative ([tex]\(\Delta < 0\)[/tex]), the solutions will be complex numbers.
Now, let's find the real part of the solutions:
[tex]\[
\text{Real part} = \frac{-b}{2a} = \frac{-(-8)}{2 \cdot 5} = \frac{8}{10} = 0.8
\][/tex]
Next, we find the imaginary part of the solutions. The magnitude of the imaginary part is calculated by:
[tex]\[
\text{Imaginary part} = \frac{\sqrt{|\Delta|}}{2a} = \frac{\sqrt{36}}{2 \cdot 5} = \frac{6}{10} = 0.6
\][/tex]
Thus, we can write the solutions as:
[tex]\[
x = \frac{0.8 \pm 0.6i}{1}
\][/tex]
To fit the form [tex]\(\frac{r - si}{t}\)[/tex] and [tex]\(\frac{r + si}{t}\)[/tex], we recognize that the coefficients are already in simplest form with [tex]\(t=1\)[/tex]. Therefore:
[tex]\[
x = \frac{8 - 6i}{10}, \quad x = \frac{8 + 6i}{10}
\][/tex]
So, the solutions are:
[tex]\[
x = \frac{8 - 6i}{10}, \quad x = \frac{8 + 6i}{10}
\][/tex]
Or to fill in the provided template:
[tex]\[
x = \frac{8 - 6i}{10}, \quad x = \frac{8 + 6i}{10}
\][/tex]
Thus, the final boxed answers will be:
[tex]\[
\boxed{8}, \boxed{6}, \boxed{10}, \boxed{8}, \boxed{10}, \boxed{10}
\][/tex]
So the complete form is:
[tex]\[
x = \frac{\boxed{8} - \boxed{6} i}{\boxed{10}}, \quad x = \frac{\boxed{8} + \boxed{6} i}{\boxed{10}}
\][/tex]
