Solve by using the quadratic equation: \( 3 x^{2}-8 x+4=0 \)

To solve the quadratic equation \( 3x^{2} - 8x + 4 = 0 \) using the quadratic formula, we start with the formula itself: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 3 \), \( b = -8 \), and \( c = 4 \). First, we calculate the discriminant (\( b^2 - 4ac \)): \[ b^2 - 4ac = (-8)^2 - 4 \cdot 3 \cdot 4 = 64 - 48 = 16 \] Now we substitute back into the quadratic formula: \[ x = \frac{-(-8) \pm \sqrt{16}}{2 \cdot 3} = \frac{8 \pm 4}{6} \] This gives us two potential solutions: 1. \( x = \frac{8 + 4}{6} = \frac{12}{6} = 2 \) 2. \( x = \frac{8 - 4}{6} = \frac{4}{6} = \frac{2}{3} \) Thus, the solutions to the equation \( 3x^2 - 8x + 4 = 0 \) are \( x = 2 \) and \( x = \frac{2}{3} \). These results highlight that the quadratic equation can yield two different roots, which is a fascinating feature of quadratic functions that can play a significant role in various applications, such as finding the points of intersection in geometry, or analyzing projectile motion in physics!