To solve the quadratic equation [tex]\( 5x^2 - 8x + 5 = 0 \)[/tex] using the quadratic formula, we follow these steps:
1. Identify the coefficients: For the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex], the coefficients are [tex]\( a = 5 \)[/tex], [tex]\( b = -8 \)[/tex], and [tex]\( c = 5 \)[/tex].
2. Calculate the discriminant: The discriminant of a quadratic equation is given by [tex]\( b^2 - 4ac \)[/tex].
[tex]\[
\text{Discriminant} = (-8)^2 - 4 \cdot 5 \cdot 5 = 64 - 100 = -36
\][/tex]
Since the discriminant is negative, the solutions will be complex numbers.
3. Calculate the real and imaginary parts:
The real part of the roots is given by [tex]\( \frac{-b}{2a} \)[/tex].
[tex]\[
\text{Real part} = \frac{-(-8)}{2 \cdot 5} = \frac{8}{10} = 0.8
\][/tex]
The imaginary part is given by [tex]\( \frac{\sqrt{|\text{discriminant}|}}{2a} \)[/tex].
[tex]\[
\text{Imaginary part} = \frac{\sqrt{36}}{2 \cdot 5} = \frac{6}{10} = 0.6
\][/tex]
4. Construct the solutions: The solutions in the form [tex]\( x = \frac{r \pm s i}{t} \)[/tex]:
[tex]\[
\text{Here, } r = 0.8, s = 0.6, t = 10
\][/tex]
Hence, the two roots are:
[tex]\[
x = \frac{0.8 - 0.6i}{10} \quad \text{and} \quad x = \frac{0.8 + 0.6i}{10}
\][/tex]
5. Convert to simplest fractional form:
Simplify the fractions:
[tex]\[
\frac{0.8}{10} = 0.08 \quad \text{and} \quad \frac{0.6}{10} = 0.06
\][/tex]
Therefore, the solutions are:
[tex]\[
x = 0.08 - 0.06i \quad \text{and} \quad x = 0.08 + 0.06i
\][/tex]
In the requested form where [tex]\( r, s \)[/tex], and [tex]\( t \)[/tex] should be integers:
Given the specific fraction and integer representation:
[tex]\[
x = \frac{0.8 - 0.6i}{10} \quad \text{and} \quad x = \frac{0.8 + 0.6i}{10}
\][/tex]
This simplifies to:
[tex]\[
x = \frac{8 - 6i}{100} \quad \text{and} \quad x = \frac{8 + 6i}{100}
\][/tex]
Thus, the solutions are:
[tex]\[
x = \frac{8 - 6i}{100}, x = \frac{8 + 6i}{100}
\][/tex]
So, the final answer in the required form is:
[tex]\[
x = \frac{8 - 6i}{100}, x = \frac{8 + 6i}{100}
\][/tex]
