Find the Roots (Zeros) 5x^2-5x+1

$5 x^{2} - 5 x + 1$

Step 1

Set $5 x^{2} - 5 x + 1$ equal to $0$ .

$5 x^{2} - 5 x + 1 = 0$

Step 2

Solve for $x$ .

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Substitute the values $a = 5$ , $b = - 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (5 \cdot 1)}}{2 \cdot 5}$

Simplify.

Simplify the numerator.

Raise $- 5$ to the power of $2$ .

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 5 \cdot 1}}{2 \cdot 5}$

Multiply $- 4 \cdot 5 \cdot 1$ .

Multiply $- 4$ by $5$ .

$x = \frac{5 \pm \sqrt{25 - 20 \cdot 1}}{2 \cdot 5}$

Multiply $- 20$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 20}}{2 \cdot 5}$

Subtract $20$ from $25$ .

$x = \frac{5 \pm \sqrt{5}}{2 \cdot 5}$

Multiply $2$ by $5$ .

$x = \frac{5 \pm \sqrt{5}}{10}$

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{5}}{10}, \frac{5 - \sqrt{5}}{10}$

$x = \frac{5 \pm \sqrt{5}}{10}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{5}}{10}$

Decimal Form:

$x = 0.72360679 \dots, 0.27639320 \dots$

Step 4