Solve $2x^2 + 3x - 2 = 0$.

How we read your problem

A quadratic equation in x: solve with the quadratic formula.

Step-by-step solution

1. 1

   Start with the equation

   $$2~{ x}^{2}+3~ x-2 = 0$$
2. 2

   Move every term to one side to get standard form

   $$2x^{2} + 3x - 2 = 0$$
3. 3

   Compute the discriminant

   $$\Delta = b^2 - 4ac = 3^2 - 4 \cdot 2 \cdot (-2) = 25$$
4. 4

   Δ > 0, so there are two distinct real roots

5. 5

   Apply the quadratic formula

   $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-3 \pm \sqrt{25}}{4} = \frac{-3 \pm 5}{4}$$
6. 6

   Simplify each root

   $$x_1 = \frac{-3 - 5}{4} = -2,\quad x_2 = \frac{-3 + 5}{4} = \frac{1}{2}$$
7. 7

   Factored form

   $$2x^2 + 3x - 2 = (x + 2)(2x - 1)$$

Formulas used

• Quadratic formula

  $$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

• Discriminant

  $$\Delta = b^2 - 4ac$$

• Vertex

  $$\left(-\frac{b}{2a},\ c - \frac{b^2}{4a}\right)$$

The quadratic formula gives every solution of ax² + bx + c = 0.

Verification

• x = -2: left side = 0, right side = 0 ✓
• x = 0.5: left side = 0, right side = 0 ✓