The graph of y = 2x² + 2x-1 is shown.
Use the graph to find estimates for
the solutions of
i) 2x²+2x-1=0


To find the estimates for the solutions of the equation \( 2x^2 + 2x - 1 = 0 \) using the graph of \( y = 2x^2 + 2x - 1 \), you need to identify the points where the graph intersects the x-axis. These points represent the solutions to the equation.

Here’s the step-by-step process:

1. **Understand the equation**: The equation \( 2x^2 + 2x - 1 = 0 \) is a quadratic equation. The solutions to this equation are the x-values where the graph intersects the x-axis (where \( y = 0 \)).

2. **Look at the graph**: On the graph of \( y = 2x^2 + 2x - 1 \), the x-axis is where \( y = 0 \). The x-values where the curve crosses the x-axis are the solutions to the equation.

3. **Estimate the solutions**: Find the x-coordinates of the points where the graph intersects the x-axis. These x-values are the approximate solutions to the equation \( 2x^2 + 2x - 1 = 0 \).

### Example Estimate:
If the graph intersects the x-axis at points around \( x = -1.5 \) and \( x = 0.5 \), then these are the approximate solutions to the equation.

So, the solutions are approximately:
- \( x \approx -1.5 \)
- \( x \approx 0.5 \)

Would you like to go into more detail or have any questions?

Here are 5 related questions to further explore the topic:
1. How would you verify the solutions algebraically?
2. What is the significance of the vertex in the graph of a quadratic equation?
3. How do the solutions of \( 2x^2 + 2x - 1 = 0 \) relate to the discriminant of the quadratic equation?
4. Can you find the exact solutions using the quadratic formula?
5. How does the graph of \( y = 2x^2 + 2x - 1 \) compare to the graph of \( y = x^2 + x - 1 \)?

**Tip:** When working with graphs, always check the x-intercepts to find the roots of the equation visually.

Learn how to estimate solutions of the quadratic equation 2x^2 + 2x - 1 = 0 using its graph. This problem explores quadratic equations and graph interpretation, providing visual solutions to algebraic problems. Suitable for high school students studying quadratic functions.

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