Solve Using the Quadratic Formula 6x^2-9x+1=x

$6 x^{2} - 9 x + 1 = x$

Step 1

Move all terms to the left side of the equation and simplify.

Subtract $x$ from both sides of the equation.

$6 x^{2} - 9 x + 1 - x = 0$

Subtract $x$ from $- 9 x$ .

$6 x^{2} - 10 x + 1 = 0$

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 $a = 6$ , $b = - 10$ , and $c = 1$ into the quadratic formula and solve for $x$ .

$\frac{10 \pm \sqrt{{(- 10)}^{2} - 4 \cdot (6 \cdot 1)}}{2 \cdot 6}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 10$ to the power of $2$ .

$x = \frac{10 \pm \sqrt{100 - 4 \cdot 6 \cdot 1}}{2 \cdot 6}$

Multiply $- 4 \cdot 6 \cdot 1$ .

Multiply $- 4$ by $6$ .

$x = \frac{10 \pm \sqrt{100 - 24 \cdot 1}}{2 \cdot 6}$

Multiply $- 24$ by $1$ .

$x = \frac{10 \pm \sqrt{100 - 24}}{2 \cdot 6}$

Subtract $24$ from $100$ .

$x = \frac{10 \pm \sqrt{76}}{2 \cdot 6}$

Rewrite $76$ as $2^{2} \cdot 19$ .

Factor $4$ out of $76$ .

$x = \frac{10 \pm \sqrt{4 (19)}}{2 \cdot 6}$

Rewrite $4$ as $2^{2}$ .

$x = \frac{10 \pm \sqrt{2^{2} \cdot 19}}{2 \cdot 6}$

Pull terms out from under the radical.

$x = \frac{10 \pm 2 \sqrt{19}}{2 \cdot 6}$

Multiply $2$ by $6$ .

$x = \frac{10 \pm 2 \sqrt{19}}{12}$

Simplify $\frac{10 \pm 2 \sqrt{19}}{12}$ .

$x = \frac{5 \pm \sqrt{19}}{6}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{19}}{6}$

Decimal Form:

$x = 1.55981649 \dots, 0.10685017 \dots$