Solve using the Quadratic Formula. \( \begin{array}{l}6 x^{2}+7 x-5=0 \\ x=\frac{-7 \pm i \sqrt{71}}{12} \\ x=-\frac{1}{2} \text { and } x=\frac{5}{3} \\ x=\frac{-7 \pm i \sqrt{71}}{2} \\ x=\frac{1}{2} \text { and } x=-\frac{5}{3}\end{array} \)

To solve the quadratic equation \(6x^2 + 7x - 5 = 0\) using the Quadratic Formula, we can follow the formula itself: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \(a = 6\), \(b = 7\), and \(c = -5\). Plugging these values into the formula, we first calculate the discriminant \(b^2 - 4ac\): \[ b^2 - 4ac = 7^2 - 4(6)(-5) = 49 + 120 = 169 \] Since the discriminant is positive, we will have two real roots. Now, substituting into the Quadratic Formula gives us: \[ x = \frac{-7 \pm \sqrt{169}}{2(6)} = \frac{-7 \pm 13}{12} \] Calculating the two possible solutions: 1. \(x = \frac{-7 + 13}{12} = \frac{6}{12} = \frac{1}{2}\) 2. \(x = \frac{-7 - 13}{12} = \frac{-20}{12} = -\frac{5}{3}\) So, the solutions are \(x = \frac{1}{2}\) and \(x = -\frac{5}{3}\). Now, for our additional fun facts: Many famous mathematicians, like Isaac Newton and Carl Friedrich Gauss, made significant contributions to understanding quadratic equations and their solutions. Quadratic equations not only appear in algebra but are also crucial in physics and engineering, helping to model projectile motion and optimize designs, showing that math really can launch you into new heights! When using the Quadratic Formula, one common mistake is miscalculating the discriminant or the square root. Always double-check your calculations and the signs! It's also essential to remember to simplify your results thoroughly—sometimes that extra step can help uncover mistakes, leading you to the correct answers much more easily.