Use a graphing utility to approximate the real solution(s), if any, of the following equation. All solution(s) lie between -10 and 10. Round to two decimal places. 𝑥􏰂 − 3𝑥 + 5.37 = 0

To approximate the real solutions of the equation \( x^2 - 3x + 5.37 = 0 \) using a graphing utility, we can proceed as follows:

1. **Rewrite the Equation:** We want to graph the function \( f(x) = x^2 - 3x + 5.37 \) and determine where it intersects the x-axis (i.e., where \( f(x) = 0 \)).

2. **Graph the Function:** Plot the function \( f(x) = x^2 - 3x + 5.37 \) within the domain \([-10, 10]\).

3. **Identify the Intersection Points:** Look for the x-values where the graph intersects the x-axis.

Let's analyze it:

### Graph Analysis
The equation is a quadratic equation, and the graph of \( f(x) = x^2 - 3x + 5.37 \) will be a parabola. To find where this parabola intersects the x-axis, we would normally look for the x-values where \( f(x) = 0 \).

### Discriminant Check
The discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) is given by \( \Delta = b^2 - 4ac \). The nature of the roots depends on the discriminant:
- If \( \Delta > 0 \), the equation has two distinct real roots.
- If \( \Delta = 0 \), the equation has exactly one real root.
- If \( \Delta < 0 \), the equation has no real roots.

For \( x^2 - 3x + 5.37 = 0 \):
- \( a = 1 \)
- \( b = -3 \)
- \( c = 5.37 \)

The discriminant is:
\[
\Delta = (-3)^2 - 4(1)(5.37) = 9 - 21.48 = -12.48
\]

Since the discriminant is negative (\( \Delta = -12.48 \)), the equation has no real roots. The graph of the function \( f(x) = x^2 - 3x + 5.37 \) will not intersect the x-axis, confirming that there are no real solutions to the equation.

### Conclusion
There are no real solutions for the equation \( x^2 - 3x + 5.37 = 0 \) within the interval \([-10, 10]\).

Would you like to explore this further, or do you have any other questions?

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**Relative Questions:**
1. How does the discriminant determine the number of real solutions of a quadratic equation?
2. What happens to the graph of a quadratic equation if the discriminant is negative?
3. Can you describe how to graph a quadratic function using its vertex and axis of symmetry?
4. What is the significance of the vertex in the graph of a quadratic equation?
5. How would the solution change if the constant term in the quadratic equation were different?

**Tip:** When dealing with quadratic equations, always check the discriminant first to determine whether real solutions exist before attempting to solve the equation graphically or algebraically.

Use a graphing utility to approximate the real solution(s), if any, of the following equation. All solution(s) lie between -10 and 10. Round to two decimal places. 𝑥^2 − 3𝑥 + 5.37 = 0

Explore how to graph the quadratic equation x^2 - 3x + 5.37 and determine if there are any real solutions between -10 and 10. This problem involves key algebraic concepts, including the discriminant and quadratic equations, making it ideal for students in Grades 9-10.

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