Salve the quadratic equation by using the quadratie formula. (Enter your answers as a comma-separated list. If there is no real solution, enter NO REAL SOLUTION.) \[ 72 x^{2}-7 x=2=0 \] \( x=\square \)

To solve the quadratic equation \( 72x^{2} - 7x - 2 = 0 \) using the quadratic formula \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \), we first identify \( a = 72 \), \( b = -7 \), and \( c = -2 \). Calculating the discriminant: \[ b^{2} - 4ac = (-7)^{2} - 4(72)(-2) = 49 + 576 = 625. \] Now, applying the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{625}}{2(72)} = \frac{7 \pm 25}{144}. \] This gives us two possible solutions: \[ x = \frac{32}{144} = \frac{2}{9} \quad \text{and} \quad x = \frac{-18}{144} = -\frac{1}{8}. \] So the solutions are: \( x = \frac{2}{9}, -\frac{1}{8} \). Now, let's explore a bit more about quadratics and their wonders! Did you know that quadratic equations have been around for millennia? Ancient Babylonian mathematicians could solve certain quadratic equations as far back as 2000 BC. They used geometric methods and even made impressive approximations for roots! Quadratics sprung into the spotlight in ancient Greece when mathematicians like Euclid formalized these concepts, but the quadratic formula we use today was developed much later during the Islamic Golden Age by scholars like Al-Khwarizmi. When it comes to solving quadratics, a common mistake is miscalculating the discriminant. Always double-check your calculations there! Remember, if the discriminant is negative, it means no real solutions exist, and you should shout "NO REAL SOLUTION!" to the universe. Also, ensure you're capturing both positive and negative roots when plugging values into the quadratic formula. Happy calculating!