Step-by-Step Factoring: Find Real Solutions for x² + 17x = -72

To solve a quadratic equation, we first need to organize it into standard form.The standard form of a quadratic equation is ax squared plus bx plus c equals zero.Let's rearrange our equation to match this form. First, we'll move all terms to the left side of the equation.Now we can identify each coefficient in our equation.The coefficient 'a' is the number in front of x squared. In this case, a equals 1.The coefficient 'b' is the number in front of x. Here, b equals 17.And 'c' is the constant term, which is 72.Organizing our equation this way is crucial for the factoring process that follows.Remember these key points about setting up a quadratic equation.With our equation properly organized, we're ready to move on to finding factor pairs.To factor this quadratic expression, we need to find two numbers with special properties.First, these numbers must multiply to give a times c, which is 72 in our case.And second, when we add these numbers together, they must give us b, which is 17.Let's systematically list out all the factor pairs of 72.Looking at all these pairs, we can see that 8 and 9 are special because they multiply to give 72 AND add to give 17.Let's verify that 8 and 9 are indeed our numbers. First, they multiply to give 72.And when we add them together, we get 17, exactly what we need!Now that we have our factors, let's write the equation in factored form.Setting this equal to zero gives us our standard factored form.By the zero product property, if a product of factors equals zero, then at least one of those factors must be zero.This means we can split our equation into two separate equations.Let's solve each equation. For the first factor, subtracting 8 from both sides gives us x equals negative 8.Similarly, for the second factor, subtracting 9 from both sides gives us x equals negative 9.Let's verify these solutions by plugging them back into the original equation.For x equals negative 8, let's substitute this value into our original equation.Similarly, for x equals negative 9, let's verify this solution as well.Both solutions check out, confirming our factoring and solving process was correct.