\( 7 x^{2}+3 x=24 \quad \) (correct to two

To solve the equation \( 7x^2 + 3x - 24 = 0 \), we can apply the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) where \( a = 7 \), \( b = 3 \), and \( c = -24 \). Plugging these values in gives us: 1. Calculate the discriminant: \( b^2 - 4ac = 3^2 - 4(7)(-24) = 9 + 672 = 681 \) 2. Now use the quadratic formula: \( x = \frac{-3 \pm \sqrt{681}}{2 \times 7} \) Calculating the two possible \( x \) values: 1. \( x_1 = \frac{-3 + \sqrt{681}}{14} \approx 1.843 \) 2. \( x_2 = \frac{-3 - \sqrt{681}}{14} \approx -1.843 \) Thus, the solutions to the equation, correct to two decimal places, are approximately \( 1.84 \) and \( -1.84 \). Many people first encounter quadratic equations in high school math, but their applications go far beyond the classroom! From calculating projectile motion in sports like baseball and basketball to optimizing profit in business models, the quadratic equation is everywhere! It’s like the Swiss Army knife of mathematics – versatile and incredibly useful in real-life problem solving. When dealing with quadratic equations, a common mistake is to miscalculate the discriminant or the values using the quadratic formula. It’s crucial to double-check the signs of all terms. For instance, forgetting to change the sign when moving terms around can land you in the wrong solution zone. Keep a keen eye on your math, and don’t be afraid to recheck your calculations!