Step-by-Step: Completing the Square Made Simple

To complete the square, we start with a quadratic equation in standard form.For this method, we require that the coefficient of x squared, which we call a, equals 1.Let's identify each term in our quadratic expression.Let's look at an example: x squared plus six x plus five equals zero.In this equation, the coefficient of x squared is 1, the coefficient of x is 6, and our constant term is 5.To prepare for completing the square, we move all terms with x to the left side and the constant term to the right side.Moving the constant term to the right side gives us x squared plus six x equals negative five.This rearrangement creates the foundation for completing the square in our next steps.Starting with our equation from the previous step:To create a perfect square, we first identify the coefficient of x, which is 6.We divide this coefficient by 2, giving us 3.Next, we square this result. Three squared equals 9.We add this number, 9, to both sides of our equation to maintain equality.Simplifying the right side, negative 5 plus 9 equals 4.The left side of our equation is now a perfect square trinomial.Let's review the steps we took to create this perfect square.Now that we have our perfect square trinomial, we're ready for the next step.Now that we've added nine to both sides, let's look at our equation.The left side of our equation, x squared plus six x plus nine, is a perfect square trinomial. We can rewrite this as x plus three squared.This gives us a simpler equation: x plus three squared equals four.To solve this, we take the square root of both sides. Remember, when we take the square root, we need both positive and negative values.This gives us x plus three equals plus or minus two. Now we can solve for x by subtracting three from both sides.Let's verify our solutions. When we plug x equals negative one back into the original equation, we get zero. Similarly, when we use x equals negative five, we also get zero.Therefore, our equation x squared plus six x plus five equals zero has two solutions: x equals negative one and x equals negative five.