How would you apply the quadratic formula to 2x^2 + 4x - 6 = 0?

Answers:

• To solve the quadratic equation ( 2x^2 + 4x - 6 = 0 ) using the quadratic formula, we first need to identify the coefficients ( a ), ( b ), and ( c ) from the standard form of a quadratic equation ( ax^2 + bx + c = 0 ).

  Here, we can identify:

  • ( a = 2 )
  • ( b = 4 )
  • ( c = -6 )

  The quadratic formula is given by:

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

  Now, we can substitute the values of ( a ), ( b ), and ( c ) into the formula:

  1. Calculate ( b^2 - 4ac ):

  [ b^2 = 4^2 = 16 ] [ 4ac = 4 \cdot 2 \cdot (-6) = -48 ] [ b^2 - 4ac = 16 - (-48) = 16 + 48 = 64 ]

  2. Now that we have ( b^2 - 4ac = 64 ), we can substitute into the quadratic formula:

  [ x = \frac{-4 \pm \sqrt{64}}{2 \cdot 2} ]

  3. Calculate ( \sqrt{64} ):

  [ \sqrt{64} = 8 ]

  4. Substitute back into the formula:

  [ x = \frac{-4 \pm 8}{4} ]

  This gives us two possible solutions:

  Solution 1:

  [ x = \frac{-4 + 8}{4} = \frac{4}{4} = 1 ]

  Solution 2:

  [ x = \frac{-4 - 8}{4} = \frac{-12}{4} = -3 ]

  Therefore, the solutions to the equation ( 2x^2 + 4x - 6 = 0 ) are:

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