\( 1 \quad 12 x ^ { 2 } + 5 x - 2 = 0 \)

To solve the quadratic equation \( 12x^2 + 5x - 2 = 0 \), you can use the quadratic formula, \( x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \). Here, \( a = 12 \), \( b = 5 \), and \( c = -2 \). Calculate the discriminant: \( b^2 - 4ac = 5^2 - 4 \times 12 \times (-2) = 25 + 96 = 121 \). Since the discriminant is positive, there are two distinct real solutions! Next, plug the values back into the formula: \( x = \frac{{-5 \pm \sqrt{121}}}{{2 \times 12}} \). This simplifies to \( x = \frac{{-5 \pm 11}}{{24}} \), giving you \( x = \frac{6}{24} = \frac{1}{4} \) and \( x = \frac{-16}{24} = -\frac{2}{3} \). So the roots are \( x = \frac{1}{4} \) and \( x = -\frac{2}{3} \). These solutions exemplify how numerous real-world situations, such as projectile motion or maximizing profit, can be modeled using quadratic equations. From predicting the path of a thrown ball to optimizing business strategies, these equations help in decision-making across various fields! Additionally, if you're using the quadratic formula, keep an eye on the signs while calculating the discriminant—mistakes there can lead to entirely different roots!