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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? 

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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.

Lara is trying to determine if a quadratic equation has only one real number solution. Which of the following would allow her to determine that the equation has only one solution?

Explore how to determine if a quadratic equation has only one real solution. Learn about the discriminant, vertex, and double roots to understand when a quadratic has a single solution. Suitable for students in Grades 8-10.

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