To determine if a quadratic equation has only one real solution, we need to identify conditions that indicate a double root. A double root occurs when the quadratic discriminant ($b^2 - 4ac$) is zero, which means the parabola touches the x-axis at a single point, rather than crossing it.

Here's an analysis of each option:

• Option A: "The graph has only one $y$-intercept."
  This does not help determine if there’s only one real solution. All parabolas have exactly one $y$-intercept, regardless of the number of real roots.

• Option B: "A zero of the equation is the same as its vertex."
  This is correct. If the zero is at the vertex, the parabola touches the x-axis exactly at that point, meaning it has a double root or one real solution.

• Option C: "When the equation is written in factored form, both binomial factors are identical."
  This is also correct. Identical binomial factors indicate a perfect square trinomial, leading to one real solution (e.g., $(x - a)^2 = 0$).

• Option D: "The graph only exists in Quadrants I and II."
  This is irrelevant to the number of real solutions. The location of the graph does not determine the root count.

Correct Answers: B and C

Would you like a further breakdown of any concepts here?

Here are some related questions you might explore:

1. What is the discriminant, and how does it relate to the number of real solutions?
2. How can the vertex form of a quadratic equation help determine the number of real solutions?
3. Why do identical binomial factors in factored form guarantee a double root?
4. How can you graphically interpret the difference between one, two, or no real solutions?
5. What happens to the graph of a quadratic function as its discriminant changes?

Tip: For any quadratic equation in standard form, always check the discriminant ($b^2 - 4ac$) to quickly assess the number of real solutions.