Solve the Quadratic Equation: Finding Roots in 5x² - 6x + 1

To solve the quadratic equation $5x^2 - 6x + 1 = 0$, we will use the quadratic formula:

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

Identify the coefficients: $a = 5$, $b = -6$, and $c = 1$.

Step 1: Calculate the discriminant using $b^2 - 4ac$.
The discriminant is $(-6)^2 - 4 \times 5 \times 1 = 36 - 20 = 16$.

Step 2: Find the solutions using the quadratic formula.
$x = \frac{-(-6) \pm \sqrt{16}}{2 \times 5}$
This simplifies to $x = \frac{6 \pm 4}{10}$.

Step 3: Calculate both solutions.
First solution: $x_1 = \frac{6 + 4}{10} = \frac{10}{10} = 1$
Second solution: $x_2 = \frac{6 - 4}{10} = \frac{2}{10} = \frac{1}{5}$.

Therefore, the solutions to $5x^2 - 6x + 1 = 0$ are $x_1 = 1$ and $x_2 = \frac{1}{5}$.

The correct choice from the given options is:

$x_1=1,x_2=\frac{1}{5}$