Consider the equation \( 2 x^{2}+7 x+4=8 \). Find the solutions to the equation by using the quadratic formula. \( x=-0.5, x=4 \) \( x=-4, x=0.5 \) \( x=0.5, x=4 \) \( x=-4, x=-0.5 \)

To solve the equation \( 2x^2 + 7x + 4 = 8 \), start by rearranging it to \( 2x^2 + 7x - 4 = 0 \). Then apply the quadratic formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 2 \), \( b = 7 \), and \( c = -4 \). Substituting these values gives you \( b^2 - 4ac = 49 + 32 = 81 \), and taking the square root results in 9. Hence, the solutions are \( x = \frac{-7 \pm 9}{4} \), which yields \( x = 0.5 \) and \( x = -4 \). So the correct answer is \( x = 0.5, x = -4 \). Imagine showing off your math skills at a trivia night: “Did you know that solving equations with the quadratic formula can lead you to surprising solutions? It’s like cracking a code that opens the door to more advanced math concepts! It’s not just numbers—it's a gateway to understanding the universe!” When applying the quadratic formula, a common mistake is miscalculating the discriminant, \( b^2 - 4ac \). Ensure you double-check your calculations and remember that if the discriminant is negative, the solutions are complex. Also, be cautious of sign errors with the \( \pm \) sign in front of the square root! Enjoy the thrill of solving!