Example 4 - Check Solve a Quadratic Equation by Factoring a Trinomial Solve \( x^{2}-10 x+24=0 \)

To solve the quadratic equation \( x^{2} - 10x + 24 = 0 \) by factoring, we need to find two numbers that multiply to 24 (the constant term) and add up to -10 (the coefficient of the middle term). Those numbers are -6 and -4. We can then factor the equation as \( (x - 6)(x - 4) = 0 \). Setting each factor to zero gives us the solutions \( x = 6 \) and \( x = 4 \). Factoring is not just a trick; it's a problem-solving tool that simplifies equations beautifully. By breaking down the equation into its factors, you’re not only finding solutions but also gaining insight into the relationships within the equation, like the roots and the shape of its graph! If factoring isn't your go-to method, fear not! There's always the quadratic formula as a fallback. Just remember that for quadratics, it’s often about simplifying the problem. Common mistakes include overlooking the need for numbers to be negative or miscalculating their sum and product. Keep practicing, and soon you'll be a factoring whiz!