Final answer:
To convert a standard form quadratic equation into both factored and vertex forms, one must find the roots using factoring or the quadratic formula, and then apply those roots to obtain the factored form, and use the vertex formula, completing the square, or the roots' average to find the vertex to write the vertex form.
The correct answer is D.
Step-by-step explanation:
The process of converting standard form into factored and vertex forms involves using algebraic techniques such as completing the square, factoring, and applying the quadratic formula. Given a quadratic equation of the form ax² + bx + c = 0, we can express it in factored form as f(x) = a(x - r1)(x - r2), where r1 and r2 are the roots or solutions to the equation. The vertex form is f(x) = a(x - h)² + k, where the vertex of the parabola is at the point (h, k).
To find the roots (r1 and r2) to formulate the factored form, we typically need to either factor directly if it's simple enough or apply the quadratic formula: x = (-b ± √(b² - 4ac)) / (2a). Once we have the roots, we convert this to factored form by plugging in the values of r1 and r2. To convert to vertex form, we can either complete the square or use the fact that the vertex h is the average of r1 and r2, and k can be found by plugging h into the original equation.