Find All Complex Solutions 2x^2=11x-10

$2 x^{2} = 11 x - 10$

Step 1

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

Subtract $11 x$ from both sides of the equation.

$2 x^{2} - 11 x = - 10$

Add $10$ to both sides of the equation.

$2 x^{2} - 11 x + 10 = 0$

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 $a = 2$ , $b = - 11$ , and $c = 10$ into the quadratic formula and solve for $x$ .

$\frac{11 \pm \sqrt{{(- 11)}^{2} - 4 \cdot (2 \cdot 10)}}{2 \cdot 2}$

Step 4

Simplify.

Simplify the numerator.

Raise $- 11$ to the power of $2$ .

$x = \frac{11 \pm \sqrt{121 - 4 \cdot 2 \cdot 10}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 10$ .

Multiply $- 4$ by $2$ .

$x = \frac{11 \pm \sqrt{121 - 8 \cdot 10}}{2 \cdot 2}$

Multiply $- 8$ by $10$ .

$x = \frac{11 \pm \sqrt{121 - 80}}{2 \cdot 2}$

Subtract $80$ from $121$ .

$x = \frac{11 \pm \sqrt{41}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{11 \pm \sqrt{41}}{4}$

Step 5

The final answer is the combination of both solutions.

$x = \frac{11 + \sqrt{41}}{4}, \frac{11 - \sqrt{41}}{4}$

Step 6

The result can be shown in multiple forms.

Exact Form:

$x = \frac{11 + \sqrt{41}}{4}, \frac{11 - \sqrt{41}}{4}$

Decimal Form:

$x = 4.35078105 \dots, 1.14921894 \dots$