Solve for x 3x^2-5x+1=0

$3 x^{2} - 5 x + 1 = 0$

Step 1

Use the quadratic formula to find the solutions.

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

Step 2

Substitute the values $a = 3$ , $b = - 5$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{5 \pm \sqrt{{(- 5)}^{2} - 4 \cdot (3 \cdot 1)}}{2 \cdot 3}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{5 \pm \sqrt{25 - 4 \cdot 3 \cdot 1}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 1$ .

Multiply $- 4$ by $3$ .

$x = \frac{5 \pm \sqrt{25 - 12 \cdot 1}}{2 \cdot 3}$

Multiply $- 12$ by $1$ .

$x = \frac{5 \pm \sqrt{25 - 12}}{2 \cdot 3}$

Subtract $12$ from $25$ .

$x = \frac{5 \pm \sqrt{13}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{5 \pm \sqrt{13}}{6}$

Step 4

The final answer is the combination of both solutions.

$x = \frac{5 + \sqrt{13}}{6}, \frac{5 - \sqrt{13}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 + \sqrt{13}}{6}, \frac{5 - \sqrt{13}}{6}$

Decimal Form:

$x = 1.43425854 \dots, 0.23240812 \dots$