Lucy and Bean have 45 marbles together. After losing 5 marbles each, the product of the number 
of marbles they have now is 124. Find out how many marbles they had to start with. 

Let's define:

- \( x \) as the number of marbles Lucy had initially.
- \( y \) as the number of marbles Bean had initially.

We know two things from the problem:

1. Lucy and Bean had a total of 45 marbles together:
   \[
   x + y = 45
   \]

2. After losing 5 marbles each, the product of their remaining marbles is 124. The number of marbles they have after losing 5 is \( (x - 5) \) for Lucy and \( (y - 5) \) for Bean:
   \[
   (x - 5)(y - 5) = 124
   \]

Now we have the system of equations:
1. \( x + y = 45 \)
2. \( (x - 5)(y - 5) = 124 \)

### Step 1: Solve for \( y \)
From the first equation, solve for \( y \):
\[
y = 45 - x
\]

### Step 2: Substitute into the second equation
Substitute \( y = 45 - x \) into the second equation:
\[
(x - 5)((45 - x) - 5) = 124
\]
Simplifying:
\[
(x - 5)(40 - x) = 124
\]

### Step 3: Expand and simplify the equation
Expand the product:
\[
(x - 5)(40 - x) = 40x - x^2 - 200 + 5x = 124
\]
Simplifying further:
\[
-x^2 + 45x - 200 = 124
\]

### Step 4: Rearrange into a quadratic equation
Move all terms to one side of the equation:
\[
-x^2 + 45x - 324 = 0
\]
Multiply by -1 to simplify:
\[
x^2 - 45x + 324 = 0
\]

### Step 5: Solve the quadratic equation
We can solve the quadratic equation \( x^2 - 45x + 324 = 0 \) using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -45 \), and \( c = 324 \).

Let's calculate the discriminant:
\[
\Delta = (-45)^2 - 4(1)(324) = 2025 - 1296 = 729
\]

Now, calculate the roots:
\[
x = \frac{45 \pm \sqrt{729}}{2} = \frac{45 \pm 27}{2}
\]

Thus, the two solutions are:
\[
x = \frac{45 + 27}{2} = 36 \quad \text{or} \quad x = \frac{45 - 27}{2} = 9
\]

### Step 6: Find the corresponding values of \( y \)
For \( x = 36 \), \( y = 45 - 36 = 9 \).

For \( x = 9 \), \( y = 45 - 9 = 36 \).

### Conclusion
Lucy and Bean originally had 36 and 9 marbles, respectively.

Would you like a more detailed breakdown of any of these steps?

Here are 5 related questions to explore:

1. How can we check the solution to verify that the product of their remaining marbles is 124?
2. What would the answer be if they had lost a different number of marbles?
3. How do we solve quadratic equations using other methods like completing the square?
4. Can you express the product equation as a function and analyze its graph?
5. How would the problem change if they had a different total number of marbles initially?

**Tip**: Always verify your solution by substituting back into the original problem conditions!

Lucy and Bean have 45 marbles together. After losing 5 marbles each, the product of the number of marbles they have now is 124. Find out how many marbles they had to start with.

Learn how to solve a system of equations involving the total number of marbles Lucy and Bean had. This detailed breakdown uses algebraic concepts and the quadratic formula to find how many marbles each had initially. Suitable for Grades 9-10.

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