Solve Using the Quadratic Formula 2x^2+5x+3=0

$2 x^{2} + 5 x + 3 = 0$

Step 1

Use the quadratic formula to find the solutions.

$\frac{- b \pm \sqrt{b^{2} - 4 (a c)}}{2 a}$

Step 2

Substitute the values $a = 2$ , $b = 5$ , and $c = 3$ into the quadratic formula and solve for $x$ .

$\frac{- 5 \pm \sqrt{5^{2} - 4 \cdot (2 \cdot 3)}}{2 \cdot 2}$

Step 3

Simplify.

Simplify the numerator.

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

$x = \frac{- 5 \pm \sqrt{25 - 4 \cdot 2 \cdot 3}}{2 \cdot 2}$

Multiply $- 4 \cdot 2 \cdot 3$ .

Multiply $- 4$ by $2$ .

$x = \frac{- 5 \pm \sqrt{25 - 8 \cdot 3}}{2 \cdot 2}$

Multiply $- 8$ by $3$ .

$x = \frac{- 5 \pm \sqrt{25 - 24}}{2 \cdot 2}$

Subtract $24$ from $25$ .

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

Any root of $1$ is $1$ .

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

Multiply $2$ by $2$ .

$x = \frac{- 5 \pm 1}{4}$

Step 4

The final answer is the combination of both solutions.

$x = - 1, - \frac{3}{2}$