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The task in the image asks to determine the canonical form of the following quadratic functions:

1. \( f(x) = x^2 - 6x + 5 \)
2. \( f(x) = x^2 + 5x + 4 \)

The canonical form of a quadratic function is typically written as:

\[
f(x) = a(x - h)^2 + k
\]

where \(a\), \(h\), and \(k\) are constants, and \( (h, k) \) represents the vertex of the parabola.

### a. \( f(x) = x^2 - 6x + 5 \)

To convert this into canonical form, we complete the square:

1. Start with \( x^2 - 6x \).
2. Take half of the coefficient of \(x\), which is \( -6/2 = -3 \), and square it: \( (-3)^2 = 9 \).
3. Add and subtract \(9\) inside the equation: 
   
   \[
   f(x) = x^2 - 6x + 9 - 9 + 5
   \]
   
   \[
   f(x) = (x - 3)^2 - 4
   \]

Thus, the canonical form is:

\[
f(x) = (x - 3)^2 - 4
\]

### b. \( f(x) = x^2 + 5x + 4 \)

Similarly, we complete the square:

1. Start with \( x^2 + 5x \).
2. Take half of the coefficient of \(x\), which is \( 5/2 \), and square it: \( \left( \frac{5}{2} \right)^2 = \frac{25}{4} \).
3. Add and subtract \( \frac{25}{4} \):

   \[
   f(x) = x^2 + 5x + \frac{25}{4} - \frac{25}{4} + 4
   \]

   \[
   f(x) = \left( x + \frac{5}{2} \right)^2 - \frac{9}{4}
   \]

Thus, the canonical form is:

\[
f(x) = \left( x + \frac{5}{2} \right)^2 - \frac{9}{4}
\]

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Would you like further clarification on these steps or details about canonical form?

### Additional Questions:
1. How is completing the square useful in solving quadratic equations?
2. What is the geometrical interpretation of the vertex form of a quadratic function?
3. Can you apply the same method to cubic functions? Why or why not?
4. How can you find the vertex directly without completing the square?
5. What are the advantages of expressing a quadratic in canonical form?

### Tip:
When completing the square, always remember to balance the equation by adding and subtracting the same value to avoid changing the function's original form.

Déterminer la forme canonique des fonctions suivantes: a. f(x) = x^2 - 6x + 5 b. f(x) = x^2 + 5x + 4

Determine the canonical form of the quadratic functions f(x) = x^2 - 6x + 5 and f(x) = x^2 + 5x + 4. Learn the steps for completing the square and rewriting the equations in vertex form. Suitable for students in Grades 10-12.

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