The graph of a quadratic equation is symmetrical about X-axis, which ones of the following can't be the roots of a quadratic equation?

Options

A. A)(0,0)
B. B)`((-2)/3,2/3)`
C. C)`(2sqrt3, -2sqrt3)`
D. D)`(4+sqrt2,4-sqrt2)`

Answer

To solve the problem, we need to analyze the properties of the roots of a quadratic equation whose graph is symmetrical about the X-axis. Let's break it down step by step.

### Step 1: Understanding Symmetry about the X-axis
A quadratic equation can be expressed in the form \( ax^2 + bx + c = 0 \). The graph of this equation is symmetrical about the X-axis if the roots of the equation are equal in magnitude but opposite in sign. This means if one root is \( \alpha \), the other root must be \( -\alpha \).

### Step 2: Relationship Between Roots
If the roots are \( \alpha \) and \( -\alpha \), we can express the sum of the roots as:
\[ \alpha + (-\alpha) = 0 \]
This indicates that for the quadratic equation to be symmetrical about the X-axis, the sum of the roots must equal zero.

### Step 3: Analyzing the Options
Now, we will evaluate the provided options to see which one does not satisfy the condition that the sum of the roots equals zero.

1. **Option A**: Roots are \( r_1 \) and \( r_2 \) such that \( r_1 + r_2 = 0 \) (valid).
2. **Option B**: Roots are \( s_1 \) and \( s_2 \) such that \( s_1 + s_2 = 0 \) (valid).
3. **Option C**: Roots are \( t_1 \) and \( t_2 \) such that \( t_1 + t_2 = 0 \) (valid).
4. **Option D**: Roots are \( u_1 \) and \( u_2 \) such that \( u_1 + u_2 = 8 \) (not valid).

### Step 4: Conclusion
Since the requirement for the roots to be symmetrical about the X-axis is that their sum must be zero, we find that the fourth option, where the sum of the roots is 8, does not satisfy this condition. Therefore, this option cannot represent the roots of a quadratic equation that is symmetrical about the X-axis.

### Final Answer
The option that cannot be the roots of a quadratic equation is **Option D**.

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