Solving x² - 2x - 48 = 0: Easy Steps

A quadratic equation in standard form follows this structure:Let's understand what each letter represents:Let's look at our specific equation: x squared minus two x minus forty-eight equals zero.We can identify the coefficients:Notice how each term corresponds to our standard form:Let's organize our work clearly. First, write the original equation.Then verify it's in standard form, with all terms on one side and zero on the other.This equation is ideal for solving by factoring because the coefficients are simple whole numbers, with no fractions or decimals.Now that we've identified and organized our equation, we're ready to begin factoring.To find the factors, we need two numbers that multiply to give ac, which is negative forty-eight, and add to give b, which is negative two.Let's systematically check different factor pairs of negative forty-eight to find the ones that work.We've found our numbers! Negative eight and positive six give us the sum of negative two and product of negative forty-eight.Let's verify these numbers work. First, we check their sum.Then, we verify their product.Now we can rewrite our middle term negative two x as negative eight x plus six x.This rewriting step is crucial for the factoring by grouping method we'll use next.Now that we've rewritten our middle term, let's factor this equation.We can group the terms with common factors.Factor out x from the first two terms and 6 from the last two terms.Notice that (x minus 8) is a common factor. We can factor it out to get our final factored form.Now we can use the zero product property to solve this equation.This means either x minus 8 equals zero, or x plus 6 equals zero.Solving these equations gives us x equals 8 and x equals negative 6.Let's verify both solutions by plugging them back into our original equation.When we substitute x equals 8, we get 64 minus 16 minus 48, which equals zero.And when we substitute x equals negative 6, we get 36 plus 12 minus 48, which also equals zero.