How many times do the parabolas given by the equation `f (x) =3x ^(2) -24x +52and g (x) =x ^(2) +12x-110` intersect ?

Options

A. Never
B. Once
C. Twice
D. More than twice

Explanation

To find how many times the parabolas given by the equations \( f(x) = 3x^2 - 24x + 52 \) and \( g(x) = x^2 + 12x - 110 \) intersect, we need to set the two functions equal to each other and solve for \( x \).

### Step-by-Step Solution:

1. **Set the equations equal to each other:**
\[ 3x^2 - 24x + 52 = x^2 + 12x - 110 \]

2. **Rearrange the equation:**
Move all terms to one side of the equation:
\[ 3x^2 - 24x + 52 - x^2 - 12x + 110 = 0 \]
Simplifying this gives:
\[ 2x^2 - 36x + 162 = 0 \]

3. **Divide the entire equation by 2 to simplify:**
\[ x^2 - 18x + 81 = 0 \]

4. **Recognize the quadratic as a perfect square:**
The equation can be factored as:
\[ (x - 9)^2 = 0 \]

5. **Solve for \( x \):**
Setting the factored equation equal to zero gives:
\[ x - 9 = 0 \implies x = 9 \]

6. **Determine the number of intersections:**
Since the equation \( (x - 9)^2 = 0 \) has a double root at \( x = 9 \), this means that the two parabolas intersect at exactly one point.

### Final Answer:
The parabolas intersect at one point.

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