Solve equation by vsing the quadratic Formbla \[ 3 x^{2}+2 x=5 \]

To solve the equation \( 3x^{2} + 2x - 5 = 0 \) using the quadratic formula, we first identify \( a = 3 \), \( b = 2 \), and \( c = -5 \). The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] Plugging in the values, we first calculate the discriminant: \[ b^{2} - 4ac = 2^{2} - 4(3)(-5) = 4 + 60 = 64 \] Now, substituting back into the formula: \[ x = \frac{-2 \pm \sqrt{64}}{2 \cdot 3} = \frac{-2 \pm 8}{6} \] This gives us two solutions: 1. \( x = \frac{6}{6} = 1 \) 2. \( x = \frac{-10}{6} = -\frac{5}{3} \) So the solutions to the equation are \( x = 1 \) and \( x = -\frac{5}{3} \). You can check your answers by substituting these values back into the original equation to confirm they satisfy it! But keep in mind that quadratic equations aren't just academic exercises; they pop up in the real world more often than you'd think! For instance, projectile motion problems often rely on quadratic equations to predict the height of a thrown ball over time. You could also run into them in business when calculating profit maximization. Fun, right? Also, make sure to avoid common mistakes when using the quadratic formula! It’s easy to miscalculate the discriminant or forget to apply the plus/minus sign when finding both solutions. Always double-check your work, and keep your calculations organized!