Final answer:
To solve an equation using a completed square, one must first create a perfect square trinomial on the left side and then use the square of a binomial pattern to rewrite it as a squared binomial. Next, take the square root of both sides. To form a single fraction on the right side with a common denominator of 4a, adjust the numerator and denominator of existing fractions until they match.
Step-by-step explanation:
After creating a perfect square trinomial on the left side of an equation, the next steps typically involve using the property that a binomial squared is equal to the perfect square trinomial. This process is known as completing the square and it allows us to solve the equation by taking the square root of both sides. To ensure that the equation maintains its equality, we must add the same value to both sides of the equation to form the perfect square trinomial.
Using the square of a binomial follows the pattern (a + b)2 = a2 + 2ab + b2. This allows us to express the left side of the equation as a squared binomial. For example, if we have x2 + 6x, we can look for a number whose square when added makes a perfect square trinomial. Here, (6/2)2 = 9, so we add 9 to both sides, leading to x2 + 6x + 9.
To form a single fraction with a common denominator of 4a on the right side, we would multiply the numerator and denominator of the fractions on the right side by whatever factors are necessary to achieve 4a as the denominator. If there's already a fraction with a denominator of a, we would multiply its numerator and denominator by 4. For fractions with different denominators, we would adjust accordingly to obtain the same common denominator of 4a.