Final answer:
Completing the square for the equation 9x² - 18x + 13 involves dividing all terms by the coefficient of x² and then forming a perfect square trinomial. The result is (x - 1)² + 4, which is a simplified form that can be used for further calculations.
Step-by-step explanation:
Completing the square is a method used to turn a quadratic equation into a perfect square trinomial, making it easier to solve. For the quadratic equation 9x² - 18x + 13, we aim to rewrite it in the form of (ax + b)² = c. The process of completing the square involves finding a value that, when added and subtracted to the equation, forms a perfect square trinomial.
We start by taking the coefficient of x², which is 9, and dividing it from the x-term, so our equation now is x² - 2x (after dividing all terms by 9). Next, we take the coefficient of x, which is -2, divide it by 2, and square it, getting 1. When we add and subtract that number inside the parentheses, our equation becomes (x - 1)². However, since we need to maintain the equality, we must subtract the square of the number outside the parentheses, multiplied by 9 (the original coefficient of x²), so we subtract 9(1) from the constant term 13.
Thus, the complete square form of the quadratic equation is (x - 1)² + 4. This form can now be used to solve for x or to graph the parabola represented by the original quadratic equation.