Step 1: Read the values of $ a $, $ b $, and $ c $ from the quadratic equation: $ a $ is the number in front of $ x^2 $, $ b $ is the number in front of $ x $, $ c $ is the number at the end. In our case:

$$ a = 2, \,\, b = 3, \,\, c = -2 $$

Step 2: Plug in the values for $ a $, $ b $, and $ c $ into the quadratic formula.

$$ \begin{aligned} x_1,x_2 &= \frac{-b \pm \sqrt{b^2-4ac}}{2a} \\[1 em] x_1,x_2 &= \frac{ -3 \pm \sqrt{ 3 ^2 - 4 \cdot 2 \cdot (-2)} }{ 2 \cdot 2 } \end{aligned} $$

Step 3: Simplify expression under the square root.

$$ x_1,x_2 = \frac{ -3 \pm \sqrt{ 25 } }{ 4 } $$

Step 4: Solve for $ x $

$$ \begin{aligned} & \color{blue}{ x_1 = \frac{ -3~-~\sqrt{ 25 } }{ 4 } = -2 } \\\\ & \color{blue}{ x_2 = \frac{ -3~+~\sqrt{ 25 } }{ 4 } = \frac{ 1 }{ 2 } } \end{aligned} $$

Quadratic Equation Solver