2. \( 5 x^{2}-21 x-54=0 \)

We need to solve the quadratic equation   5x² – 21x – 54 = 0. There are several methods to solve a quadratic equation. Here, we'll show both the factoring method and the quadratic formula. ───────────────────────────── Step 1. Factoring We look for two numbers a and b such that when the quadratic is factored in the form   (ax + m)(bx + n) = 0, we get   a·b = 5 (the coefficient of x²) and m·n = –54 (the constant term), and the cross terms satisfy   a·n + b·m = –21 (the coefficient of x). Since 5 is a prime number, a likely factorization is to assign 5 to one of the binomials. Try writing the expression as   (x – 6)(5x + 9) = 0. Let’s check by expanding:   (x – 6)(5x + 9) = 5x² + 9x – 30x – 54 = 5x² – 21x – 54. Since it matches the given quadratic, the factorization is correct. Setting each factor equal to zero gives: 1) x – 6 = 0  ⟹ x = 6, 2) 5x + 9 = 0  ⟹ 5x = –9  ⟹ x = –9/5. ───────────────────────────── Step 2. Quadratic Formula Alternatively, we can use the quadratic formula, which for an equation ax² + bx + c = 0 is:   x = [–b ± √(b² – 4ac)] / (2a). Here, a = 5, b = –21, and c = –54. Plugging these in: 1. Calculate the discriminant:   Δ = b² – 4ac      = (–21)² – 4·5·(–54)      = 441 + 1080      = 1521. 2. Find the square root of Δ:   √1521 = 39       (since 39² = 1521). 3. Substitute into the formula:   x = [–(–21) ± 39] / (2·5)   x = [21 ± 39] / 10. Thus, we get two solutions:   x = (21 + 39) / 10 = 60 / 10 = 6,   x = (21 – 39) / 10 = (–18) / 10 = –9/5. ───────────────────────────── Final Answer The solutions to the equation 5x² – 21x – 54 = 0 are   x = 6  and  x = –9/5.