What are the roots of the quadratic equation 3x^2 - 12x + 9 = 0?

Answers:

• To find the roots of the quadratic equation (3x^2 - 12x + 9 = 0), we can use the quadratic formula, which is given by:

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

  In our equation, the coefficients are:

  • (a = 3)
  • (b = -12)
  • (c = 9)

  First, we calculate the discriminant (b^2 - 4ac):

  [ b^2 = (-12)^2 = 144 ] [ 4ac = 4 \cdot 3 \cdot 9 = 108 ] [ b^2 - 4ac = 144 - 108 = 36 ]

  Now, we substitute these values into the quadratic formula:

  [ x = \frac{-(-12) \pm \sqrt{36}}{2 \cdot 3} ] [ x = \frac{12 \pm 6}{6} ]

  Now, we calculate the two possible values for (x):

  1. When we use the positive root: [ x = \frac{12 + 6}{6} = \frac{18}{6} = 3 ]

  2. When we use the negative root: [ x = \frac{12 - 6}{6} = \frac{6}{6} = 1 ]

  Thus, the roots of the equation (3x^2 - 12x + 9 = 0) are:

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