The equation of a curve is given as `y=x^(2)+2-3x`.
The curve intersects the x-axis at

Options

A. Statement-I is true, Statement-II is true, Statement-II is correct explanation for statement-I
B. Statement-I is true, Statement-II is true, Statement-II is NOT a correct explanation for statement-I
C. Statement-I is true, Statement-II is false.
D. Statement-I is false and Statement-II is true.

Answer

To find the points where the curve intersects the x-axis, we need to set the equation of the curve equal to zero. The given equation of the curve is:

\[ y = x^2 + 2 - 3x \]

### Step 1: Set the equation to zero
To find the x-intercepts, we set \( y = 0 \):

\[ 0 = x^2 + 2 - 3x \]

This can be rearranged to:

\[ x^2 - 3x + 2 = 0 \]

### Step 2: Identify coefficients
In the quadratic equation \( ax^2 + bx + c = 0 \), we identify the coefficients:

- \( a = 1 \)
- \( b = -3 \)
- \( c = 2 \)

### Step 3: Use the quadratic formula
The roots of the quadratic equation can be found using the quadratic formula:

\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Substituting the values of \( a \), \( b \), and \( c \):

\[ x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4 \cdot 1 \cdot 2}}{2 \cdot 1} \]

### Step 4: Simplify the expression
Calculating the discriminant:

\[ b^2 - 4ac = 9 - 8 = 1 \]

Now substituting back into the formula:

\[ x = \frac{3 \pm \sqrt{1}}{2} \]

### Step 5: Calculate the roots
This simplifies to:

\[ x = \frac{3 \pm 1}{2} \]

Calculating the two possible values for \( x \):

1. \( x_1 = \frac{3 + 1}{2} = \frac{4}{2} = 2 \)
2. \( x_2 = \frac{3 - 1}{2} = \frac{2}{2} = 1 \)

### Step 6: Write the coordinates of intersection
The curve intersects the x-axis at the points:

- \( (2, 0) \)
- \( (1, 0) \)

### Final Answer:
The curve intersects the x-axis at the points \( (2, 0) \) and \( (1, 0) \).

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