

You said:
جد نقاط التقاطع مع محور x، 
F(x)= 4x²-3 بسط 
2x²+1 مقام

To find the points of intersection with the x-axis for a function, we need to determine where the function \( F(x) \) equals zero. This occurs when the **numerator** of the function is zero (since a fraction is zero when its numerator is zero, provided the denominator is not zero).

Given the function:

\[
F(x) = \frac{4x^2 - 3}{2x^2 + 1}
\]

### Step-by-step Solution:

1. **Set the numerator equal to zero**:  
   \[
   4x^2 - 3 = 0
   \]

2. **Solve for \( x \)**:  
   \[
   4x^2 = 3
   \]  
   \[
   x^2 = \frac{3}{4}
   \]  
   \[
   x = \pm \sqrt{\frac{3}{4}}
   \]  
   \[
   x = \pm \frac{\sqrt{3}}{2}
   \]

3. **Verify that the denominator is not zero**:  
   The denominator is:
   \[
   2x^2 + 1
   \]  
   Plugging in \( x = \pm \frac{\sqrt{3}}{2} \):

   \[
   2\left(\frac{\sqrt{3}}{2}\right)^2 + 1 = 2 \times \frac{3}{4} + 1 = \frac{3}{2} + 1 = \frac{5}{2} \neq 0
   \]

   Thus, the denominator is not zero for these values of \( x \).

### Conclusion:

The points of intersection with the x-axis are:

\[
\left( \frac{\sqrt{3}}{2}, 0 \right) \quad \text{and} \quad \left( -\frac{\sqrt{3}}{2}, 0 \right)
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you find the intersection points with the y-axis for a rational function?
2. What happens if both the numerator and denominator are zero at a point?
3. How do you find vertical asymptotes for a rational function?
4. What is the significance of holes in the graph of a rational function?
5. How can you determine if a function is continuous at a given point?

### Tip:
For rational functions, always check both the numerator and the denominator when looking for zeros or intersections to avoid division by zero issues.

جد نقاط التقاطع مع محور x، F(x)= 4x²-3 بسط 2x²+1 مقام

Learn how to find the x-axis intersection points for the rational function F(x) = (4x² - 3) / (2x² + 1) with a detailed, step-by-step solution. This problem covers rational functions, solving quadratic equations, and checking for division by zero. Suitable for students in Grades 10-12.

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