Quadratic Formula (Rational Solutions) Score: 4/10 Penalty: none Question \( 6 c^{2}-c-15=0 \) Use the quadratic formula to solve. Express your answer in simplest form. \[ \text { Show Examples } \]

To use the quadratic formula, we start with the standard form of a quadratic equation \( ax^2 + bx + c = 0 \). For the equation \( 6c^2 - c - 15 = 0 \), we identify \( a = 6 \), \( b = -1 \), and \( c = -15 \). The quadratic formula is given by: \[ c = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Plugging in the values: 1. Calculate the discriminant: \[ b^2 - 4ac = (-1)^2 - 4(6)(-15) = 1 + 360 = 361 \] 2. Now substitute back into the formula: \[ c = \frac{-(-1) \pm \sqrt{361}}{2(6)} = \frac{1 \pm 19}{12} \] 3. This creates two potential solutions: \[ c = \frac{20}{12} = \frac{5}{3} \quad \text{and} \quad c = \frac{-18}{12} = -\frac{3}{2} \] Thus, the simplest forms of the solutions are \( c = \frac{5}{3} \) and \( c = -\frac{3}{2} \). Imagine solving a real-world problem, like predicting the length of a parabolic bridge or a roller coaster. Engineers often utilize the quadratic formula to calculate key parameters, ensuring safety and optimal design. By accurately solving for points of intersection or maximum heights, they ensure structures are both functional and aesthetic—who knew math could look so good? When using the quadratic formula, a common pitfall is forgetting to carefully calculate the discriminant, leading to potential errors in finding the roots. Always double-check each step, especially signs, to avoid results that don’t align with your expectations. Another mistake can occur when simplifying the final form; ensure you’re expressing your solution in the simplest terms possible!