29. Solve: \[ 3 x^{2}-4 x-3=0 \]

To solve the quadratic equation \( 3x^2 - 4x - 3 = 0 \), you can use the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 3 \), \( b = -4 \), and \( c = -3 \). Plugging in these values: \[ b^2 - 4ac = (-4)^2 - 4 \cdot 3 \cdot (-3) = 16 + 36 = 52 \] Substituting back into the quadratic formula: \[ x = \frac{-(-4) \pm \sqrt{52}}{2 \cdot 3} = \frac{4 \pm 2\sqrt{13}}{6} = \frac{2 \pm \sqrt{13}}{3} \] Thus, the solutions are: \[ x = \frac{2 + \sqrt{13}}{3} \quad \text{and} \quad x = \frac{2 - \sqrt{13}}{3} \] These values are the roots of the quadratic equation! The fun part is that quadratic equations like this one can represent real-world scenarios, from projectile motion to financial modeling. If you threw a ball into the air, its height over time could be modeled with a quadratic equation similar to this, helping you predict when it will reach its peak and return to the ground. Also, be careful with negative signs when applying the quadratic formula! Many students trip up by miscalculating the value of \( b \) or forgetting to include the ± symbol when finding the two potential solutions. Double-checking each step can make all the difference in reaching the correct answers!