Solve Using the Quadratic Formula 2x^2=5x+1

$2 x^{2} = 5 x + 1$

Step 1

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

Subtract $5 x$ from both sides of the equation.

$2 x^{2} - 5 x = 1$

Subtract $1$ from both sides of the equation.

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

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 $a = 2$ , $b = - 5$ , and $c = - 1$ into the quadratic formula and solve for $x$ .

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

Step 4

Simplify.

Simplify the numerator.

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

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

Multiply $- 4 \cdot 2 \cdot - 1$ .

Multiply $- 4$ by $2$ .

$x = \frac{5 \pm \sqrt{25 - 8 \cdot - 1}}{2 \cdot 2}$

Multiply $- 8$ by $- 1$ .

$x = \frac{5 \pm \sqrt{25 + 8}}{2 \cdot 2}$

Add $25$ and $8$ .

$x = \frac{5 \pm \sqrt{33}}{2 \cdot 2}$

Multiply $2$ by $2$ .

$x = \frac{5 \pm \sqrt{33}}{4}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{5 \pm \sqrt{33}}{4}$

Decimal Form:

$x = 2.68614066 \dots, - 0.18614066 \dots$