Solve the following equation by factorization method a. \( 3 x^{2}-5 x-2=0 \)

To solve the equation \( 3x^{2} - 5x - 2 = 0 \) by factorization, we need to express it in the form of \( (px + q)(rx + s) = 0 \). First, we find two numbers that multiply to \( 3 \times -2 = -6 \) and add up to \( -5 \). The numbers are \( -6 \) and \( 1 \). Now, we can rewrite the middle term: \( 3x^{2} - 6x + x - 2 = 0 \). Next, we group the terms: \( (3x^{2} - 6x) + (x - 2) = 0 \). Factoring out the common factors: \( 3x(x - 2) + 1(x - 2) = 0 \). Now, we can factor out \( (x - 2) \): \( (3x + 1)(x - 2) = 0 \). Setting each factor to zero gives us: 1. \( 3x + 1 = 0 \) => \( x = -\frac{1}{3} \) 2. \( x - 2 = 0 \) => \( x = 2 \) Thus, the solutions are \( x = -\frac{1}{3} \) and \( x = 2 \). For a little fun trivia, did you know that factoring is like solving a puzzle? Every time you break down an equation, you're uncovering the hidden structure of numbers – it's just like finding the secret recipe behind a delicious dish! And if you're interested in honing your skills, consider diving into algebra-focused workbooks or online courses. They often present problems that challenge your creativity while reinforcing your foundational concepts. Plus, there's nothing like the feeling of unraveling a tricky equation!