Solve the quadratic equation 3x^2 - 5x + 2 = 0

Answers:

• To solve the quadratic equation ( 3x^2 - 5x + 2 = 0 ), we can use the quadratic formula:

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

  where ( a = 3 ), ( b = -5 ), and ( c = 2 ).

  1. Calculate the discriminant:

  [ b^2 - 4ac = (-5)^2 - 4 \times 3 \times 2 = 25 - 24 = 1 ]

  2. Since the discriminant is positive, we will have two real and distinct solutions.

  3. Substitute the values into the quadratic formula:

  [ x = \frac{-(-5) \pm \sqrt{1}}{2 \times 3} = \frac{5 \pm 1}{6} ]

  4. Calculate the two possible solutions:
  • For ( 5 + 1 ):

  [ x = \frac{6}{6} = 1 ]

  • For ( 5 - 1 ):

  [ x = \frac{4}{6} = \frac{2}{3} ]

  Thus, the solutions to the equation ( 3x^2 - 5x + 2 = 0 ) are:

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

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