Solve Using the Quadratic Formula 10x^2=2x+3

$10 x^{2} = 2 x + 3$

Step 1

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

Subtract $2 x$ from both sides of the equation.

$10 x^{2} - 2 x = 3$

Subtract $3$ from both sides of the equation.

$10 x^{2} - 2 x - 3 = 0$

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 $a = 10$ , $b = - 2$ , and $c = - 3$ into the quadratic formula and solve for $x$ .

$\frac{2 \pm \sqrt{{(- 2)}^{2} - 4 \cdot (10 \cdot - 3)}}{2 \cdot 10}$

Step 4

Simplify.

Simplify the numerator.

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

$x = \frac{2 \pm \sqrt{4 - 4 \cdot 10 \cdot - 3}}{2 \cdot 10}$

Multiply $- 4 \cdot 10 \cdot - 3$ .

Multiply $- 4$ by $10$ .

$x = \frac{2 \pm \sqrt{4 - 40 \cdot - 3}}{2 \cdot 10}$

Multiply $- 40$ by $- 3$ .

$x = \frac{2 \pm \sqrt{4 + 120}}{2 \cdot 10}$

Add $4$ and $120$ .

$x = \frac{2 \pm \sqrt{124}}{2 \cdot 10}$

Rewrite $124$ as $2^{2} \cdot 31$ .

Factor $4$ out of $124$ .

$x = \frac{2 \pm \sqrt{4 (31)}}{2 \cdot 10}$

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

$x = \frac{2 \pm \sqrt{2^{2} \cdot 31}}{2 \cdot 10}$

Pull terms out from under the radical.

$x = \frac{2 \pm 2 \sqrt{31}}{2 \cdot 10}$

Multiply $2$ by $10$ .

$x = \frac{2 \pm 2 \sqrt{31}}{20}$

Simplify $\frac{2 \pm 2 \sqrt{31}}{20}$ .

$x = \frac{1 \pm \sqrt{31}}{10}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$x = \frac{1 \pm \sqrt{31}}{10}$

Decimal Form:

$x = 0.65677643 \dots, - 0.45677643 \dots$