Solve Using the Quadratic Formula x^2=3x-1

$x^{2} = 3 x - 1$

Step 1

Move all the expressions to the left side of the equation.

Subtract $3 x$ from both sides of the equation.

$x^{2} - 3 x = - 1$

Add $1$ to both sides of the equation.

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

Step 2

Use the quadratic formula to find the solutions.

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

Step 3

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

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

Step 4

Simplify.

Simplify the numerator.

Raise $- 3$ to the power of $2$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1 \cdot 1}}{2 \cdot 1}$

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

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4 \cdot 1}}{2 \cdot 1}$

Multiply $- 4$ by $1$ .

$x = \frac{3 \pm \sqrt{9 - 4}}{2 \cdot 1}$

Subtract $4$ from $9$ .

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

Multiply $2$ by $1$ .

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

Step 5

The result can be shown in multiple forms.

Exact Form:

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

Decimal Form:

$x = 2.61803398 \dots, 0.38196601 \dots$