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The image contains a series of mathematical problems written in French. I will transcribe and solve each of the problems for you:

1. **Éliminer le radical suivant : \(\sqrt{2} + \frac{4}{\sqrt{2}}\)**

   **Solution:**
   \[
   \sqrt{2} + \frac{4}{\sqrt{2}} = \sqrt{2} + \frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \sqrt{2} + \frac{4\sqrt{2}}{2} = \sqrt{2} + 2\sqrt{2} = 3\sqrt{2}
   \]
   The simplified expression is \(3\sqrt{2}\).

2. **Réduire la fraction (la transformer en fraction irréductible) : \(\frac{1274}{1876}\)**

   **Solution:**
   To reduce the fraction \(\frac{1274}{1876}\), find the greatest common divisor (GCD) of 1274 and 1876. The GCD is 2.
   \[
   \frac{1274 \div 2}{1876 \div 2} = \frac{637}{938}
   \]
   The reduced fraction is \(\frac{637}{938}\).

3. **Quel est le PPCM de 24 et 36?**

   **Solution:**
   The PPCM (Plus Petit Commun Multiple) or LCM (Least Common Multiple) of 24 and 36 is found by the following steps:
   - Prime factorization of 24: \(2^3 \times 3\)
   - Prime factorization of 36: \(2^2 \times 3^2\)
   - LCM = \(2^3 \times 3^2 = 8 \times 9 = 72\)
   So, the PPCM is 72.

4. **Résoudre l'équation suivante par rapport à x: \(x^2 - 5x + 6 = 0\)**

   **Solution:**
   The equation is a quadratic equation, and can be factored as:
   \[
   x^2 - 5x + 6 = (x - 2)(x - 3) = 0
   \]
   The solutions are \(x = 2\) and \(x = 3\).

5. **Roger a 50 ans de plus que Sylviane. Dans 7 ans, Roger aura le double de l'âge de Sylviane. Quel est l'âge actuel de Roger ?**

   **Solution:**
   Let \( x \) be Sylviane's current age. Then, Roger's age is \( x + 50 \). In 7 years, Roger's age will be \( x + 50 + 7 = x + 57 \), and Sylviane's age will be \( x + 7 \).

   According to the problem:
   \[
   x + 57 = 2(x + 7)
   \]
   Simplifying:
   \[
   x + 57 = 2x + 14 \implies 57 - 14 = 2x - x \implies x = 43
   \]
   So, Sylviane is 43 years old, and Roger is \( 43 + 50 = 93 \) years old.

Would you like more details on any of these solutions, or do you have further questions?

Here are some related questions you might find interesting:

1. How do you find the LCM of three numbers?
2. Can you provide a different method to solve quadratic equations besides factoring?
3. What is the difference between PPCM and PGCD (Greatest Common Divisor)?
4. How would the equation change if Roger were only 40 years older than Sylviane?
5. How do you simplify expressions with radicals in the denominator?

**Tip:** When solving quadratic equations, always check if the equation can be factored before using the quadratic formula—this can save time and simplify calculations!

Éliminer le radical suivant : √2 + 4/√2; Réduire la fraction : 1274/1876; Quel est le PPCM de 24 et 36?; Résoudre l'équation : x^2 - 5x + 6 = 0; Roger a 50 ans de plus que Sylviane. Dans 7 ans, Roger aura le double de l'âge de Sylviane. Quel est l'âge actuel de Roger?

Learn how to solve problems involving radicals, fractions, LCM, and quadratic equations. This guide includes detailed solutions to problems such as simplifying √2 + 4/√2, reducing fractions, finding the LCM of numbers, solving quadratic equations, and more. Suitable for Grades 8-10.

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