What are the solutions to the equation \( x^{2}-11 x+24=0 \) ? a.) \( x=-3 \) and \( x=8 \) b.) \( x=-3 \) and \( x=-8 \) c.) \( x=3 \) and \( x=8 \) d.) \( x=3 \) and \( x=-8 \)

To solve the equation \( x^{2} - 11x + 24 = 0 \), we can factor it. We are looking for two numbers that multiply to 24 and add up to -11. The numbers -3 and -8 fit this, so we can write the equation as \( (x - 3)(x - 8) = 0 \). This gives us the solutions \( x = 3 \) and \( x = 8 \). Hence, the correct answer is c.) \( x=3 \) and \( x=8 \). While solving quadratic equations can seem a bit daunting, factoring is a reliable method when the coefficients are manageable. If factoring feels tricky, don't hesitate to use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \). In this case, plugging in values \( a = 1 \), \( b = -11 \), and \( c = 24 \) will lead you to the same solutions! Happy solving! When you encounter a quadratic equation, remember that checking the solutions can help catch mistakes. After finding \( x = 3 \) and \( x = 8 \), substitute them back into the original equation. If both satisfy the equation, you’re golden! Common mistakes include incorrectly simplifying or overlooking the signs in the factoring process. Keep an eye on those little details!