Can you solve the equation 3x^2 + 5x - 2 = 0 using the quadratic formula and provide all steps?

Answers:

• To solve the quadratic equation (3x^2 + 5x - 2 = 0) using the quadratic formula, we will follow these steps:

  1. Identify coefficients: The standard form of a quadratic equation is (ax^2 + bx + c = 0). Here, we have:

     • (a = 3)
     • (b = 5)
     • (c = -2)

  2. Write the quadratic formula: The quadratic formula is given by: [ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} ]

  3. Calculate the discriminant: First, we need to compute (b^2 - 4ac). [ b^2 = 5^2 = 25 ] [ 4ac = 4 \times 3 \times -2 = -24 ] Now, substitute these values into the discriminant: [ b^2 - 4ac = 25 - (-24) = 25 + 24 = 49 ]

  4. Substitute values into the quadratic formula: Now we can substitute (a), (b), and the discriminant into the formula: [ x = \frac{-5 \pm \sqrt{49}}{2 \times 3} ]

  5. Calculate the square root: [ \sqrt{49} = 7 ] Now substitute this back into the equation: [ x = \frac{-5 \pm 7}{6} ]

  6. Solve for the two possible values of (x): We will calculate the two cases: one with the positive sign and one with the negative sign.

     • Case 1 (using the positive sign): [ x_1 = \frac{-5 + 7}{6} = \frac{2}{6} = \frac{1}{3} ]

     • Case 2 (using the negative sign): [ x_2 = \frac{-5 - 7}{6} = \frac{-12}{6} = -2 ]

  7. Final solutions: The two solutions to the equation (3x^2 + 5x - 2 = 0) are: [ x = \frac{1}{3} \quad \text{and} \quad x = -2 ]

  Thus, the final answer is:

  [ x = \frac{1}{3} \quad \text{or} \quad x = -2 ]