https://file.aiquickdraw.com/mathbot/file/1725025462483li0mubug.jpg

To help you with the items 18 to 22 in the image, I'll first extract and analyze the relevant information from the provided image.

### Extraction from the Image:
The image shows a table with mathematical expressions and tasks related to solving or manipulating these expressions. Here is a breakdown:

1. **Item I**: \( a(x) = x^2 - 2x + 3 \)
2. **Item II**: \( b(x) = \frac{2x + 3}{x + 1} \)
3. **Item III**: \( c(x) = x^2 - 3x + 1 \)
4. **Item IV**: \( d(x) = 2x^3 - 15x - 12 \)
5. **Item V**: \( e(x) = \frac{5}{x + 2} \)

### The following items are listed:
- **Item 18**: Write down the composite function \( (d \circ e)(x) \) and simplify.
- **Item 19**: Find the roots of \( c(x) = 0 \).
- **Item 20**: Determine the points of intersection of \( a(x) \) and \( b(x) \).
- **Item 21**: Find the value of \( x \) such that \( d(x) = e(x) \).
- **Item 22**: Analyze the nature of the roots of \( d(x) = 0 \) using the discriminant.

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### Let's go step by step:

#### **Item 18**: Composite function \( (d \circ e)(x) \)
The composite function \( (d \circ e)(x) \) means \( d(e(x)) \).

Given \( e(x) = \frac{5}{x+2} \), substitute this into \( d(x) \).

\[ d\left( \frac{5}{x+2} \right) = 2\left(\frac{5}{x+2}\right)^3 - 15\left(\frac{5}{x+2}\right) - 12 \]

This will need to be simplified.

#### **Item 19**: Roots of \( c(x) = 0 \)
The function \( c(x) = x^2 - 3x + 1 \) is a quadratic equation.

To find the roots, solve:
\[ x^2 - 3x + 1 = 0 \]

Using the quadratic formula:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Where \( a = 1 \), \( b = -3 \), and \( c = 1 \).

#### **Item 20**: Points of intersection of \( a(x) \) and \( b(x) \)
Find the \( x \) such that \( a(x) = b(x) \).

\[ x^2 - 2x + 3 = \frac{2x + 3}{x + 1} \]

Cross multiply and solve for \( x \).

#### **Item 21**: \( d(x) = e(x) \)
Set \( d(x) = e(x) \) and solve for \( x \).

\[ 2x^3 - 15x - 12 = \frac{5}{x + 2} \]

Multiply both sides by \( x + 2 \) and solve for \( x \).

#### **Item 22**: Nature of the roots of \( d(x) = 0 \)
To analyze the nature of the roots of \( d(x) = 0 \), you need to find the discriminant of the cubic equation \( d(x) = 2x^3 - 15x - 12 = 0 \).

The discriminant of a cubic equation \( ax^3 + bx^2 + cx + d = 0 \) is given by a more complex formula, but it generally tells us if the equation has one real root and two complex roots, or three real roots.

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Would you like to go through these calculations in more detail, or do you need help with specific parts of any item?

Explore advanced mathematics with problems on composite functions, quadratic equations, and cubic equations. Learn how to solve composite function problems, find roots of quadratic and cubic equations, and analyze their discriminants. Suitable for advanced high school students.

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