M2 – 8M – 48 = 0…………… (1)

50% of (Z + Y + M) = 19………… (2)

Z, Y and M are positive integers such that out of three any two are same.

Let's solve the given system of equations step by step.

### Step 1: Solve the quadratic equation for \( M \)

Given:
\[ M^2 - 8M - 48 = 0 \]

This is a quadratic equation in the form of \( ax^2 + bx + c = 0 \). We can solve it using the quadratic formula:
\[ M = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Here, \( a = 1 \), \( b = -8 \), and \( c = -48 \). Plugging these values into the quadratic formula, we get:
\[ M = \frac{8 \pm \sqrt{64 + 192}}{2} \]
\[ M = \frac{8 \pm \sqrt{256}}{2} \]
\[ M = \frac{8 \pm 16}{2} \]

This gives us two solutions:
\[ M = \frac{8 + 16}{2} = 12 \]
\[ M = \frac{8 - 16}{2} = -4 \]

Since \( M \) is a positive integer, we take \( M = 12 \).

### Step 2: Solve the second equation for \( Z \) and \( Y \)

Given:
\[ 50\% \text{ of } (Z + Y + M) = 19 \]

This can be rewritten as:
\[ \frac{Z + Y + M}{2} = 19 \]
\[ Z + Y + M = 38 \]

We know \( M = 12 \), so:
\[ Z + Y + 12 = 38 \]
\[ Z + Y = 26 \]

### Step 3: Find the values of \( Z \) and \( Y \)

We are given that \( Z \), \( Y \), and \( M \) are positive integers such that any two of them are the same. Given \( M = 12 \), we need to find \( Z \) and \( Y \) such that they are positive integers and any two of the three numbers are the same.

Since \( Z \) and \( Y \) must satisfy \( Z + Y = 26 \), let's check possible values:
- If \( Z = 12 \), then \( Y = 26 - 12 = 14 \). This pair works since one of \( Z \) and \( Y \) equals \( M \).

Therefore, the solution that fits the given conditions is:
\[ Z = 12 \]
\[ Y = 14 \]

Thus, the values are:
\[ M = 12 \]
\[ Z = 12 \]
\[ Y = 14 \]

Would you like more details or have any questions?

Here are 8 related questions to expand on the previous information:
1. How would the solution change if the quadratic equation had different coefficients?
2. Can you solve another quadratic equation with different coefficients?
3. What are the general steps to solve a system of linear equations with multiple variables?
4. How do you verify if the solutions to the system of equations are correct?
5. What are the properties of quadratic equations that help in solving them?
6. Can you find the value of \( Z \) and \( Y \) if \( M \) was a different positive integer?
7. How do you solve a system of equations when the variables must be positive integers?
8. What are the different methods to solve quadratic equations?

**Tip:** Always check the conditions given in the problem statement after solving equations to ensure that the solutions meet all the criteria.

Learn how to solve a quadratic equation and a system of equations with positive integer solutions step-by-step. This problem involves applying the quadratic formula and solving for integer values that satisfy specified conditions.

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