Solve for x 3x^2-12x+11=x

$3 x^{2} - 12 x + 11 = x$

Step 1

Move all terms containing $x$ to the left side of the equation.

Subtract $x$ from both sides of the equation.

$3 x^{2} - 12 x + 11 - x = 0$

Subtract $x$ from $- 12 x$ .

$3 x^{2} - 13 x + 11 = 0$

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 $a = 3$ , $b = - 13$ , and $c = 11$ into the quadratic formula and solve for $x$ .

$\frac{13 \pm \sqrt{{(- 13)}^{2} - 4 \cdot (3 \cdot 11)}}{2 \cdot 3}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 13$ to the power of $2$ .

$x = \frac{13 \pm \sqrt{169 - 4 \cdot 3 \cdot 11}}{2 \cdot 3}$

Multiply $- 4 \cdot 3 \cdot 11$ .

Multiply $- 4$ by $3$ .

$x = \frac{13 \pm \sqrt{169 - 12 \cdot 11}}{2 \cdot 3}$

Multiply $- 12$ by $11$ .

$x = \frac{13 \pm \sqrt{169 - 132}}{2 \cdot 3}$

Subtract $132$ from $169$ .

$x = \frac{13 \pm \sqrt{37}}{2 \cdot 3}$

Multiply $2$ by $3$ .

$x = \frac{13 \pm \sqrt{37}}{6}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{13 + \sqrt{37}}{6}, \frac{13 - \sqrt{37}}{6}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{13 + \sqrt{37}}{6}, \frac{13 - \sqrt{37}}{6}$

Decimal Form:

$x = 3.18046042 \dots, 1.15287291 \dots$