Final answer:
Completing the Square is the method that requires the coefficient of (z¹⁰) to be 1 before solving a quadratic equation. Unlike Factoring and the Quadratic Formula, this method involves transforming the quadratic into a perfect square trinomial which demands the leading coefficient to be 1. The correct answer is option A .
Step-by-step explanation:
When solving a quadratic equation, it's important to understand the appropriate method to use based on the form of the equation. The question of which method requires the coefficient of (z¹⁰) to be 1 before solving a quadratic equation is asked, and the answer to this is Completing the Square. This method necessitates that the coefficient of the squared term (commonly referred to as a in the quadratic equation ax² + bx + c = 0) must be 1 in order to complete the square properly. Other methods, such as the Quadratic Formula and Factoring, do not have this requirement as they can be applied directly to the quadratic equation regardless of the coefficient.
Factoring a quadratic is often the simplest method, but it requires that the quadratic be factorable in the first place. The Quadratic Formula, which is derived from the process of completing the square, can solve any quadratic equation regardless of the coefficient and does not require simplifying the equation beforehand. Lastly, the option None of the Above is not correct as Completing the Square does require the coefficient of (z¹⁰), or the quadratic term, to be 1.
The process of Completing the Square involves taking a quadratic equation in the form of ax² + bx + c, and manipulating it so that it becomes a perfect square trinomial, (x+d)², making it much easier to solve for x. To achieve this, the equation must first be in the form x² + (b/a)x, which is why the coefficient of x² must be 1. If it is not already 1, you would divide the entire equation by a before proceeding to complete the square.
In summary, Completing the Square is distinct from other methods such as the Quadratic Formula and Factoring in that it requires the leading coefficient to be 1, which directly affects the transformation of the quadratic into a perfect square trinomial.