Let's define the present age of the father as $x$ and the present age of the son as $y$.

Step 1: Set up the equations

We are given two pieces of information:

1. The sum of their ages is 45 years: $x + y = 45$
2. Five years ago, the product of their ages was 124: $(x - 5)(y - 5) = 124$

Step 2: Expand the second equation

Now, let's expand the second equation: $(x - 5)(y - 5) = xy - 5x - 5y + 25$ Substitute $x + y = 45$ into the equation: $xy - 5(x + y) + 25 = 124$ $xy - 5(45) + 25 = 124$ $xy - 225 + 25 = 124$ $xy - 200 = 124$ $xy = 324$

Step 3: Solve the system of equations

Now we have the system of equations: $x + y = 45$ $xy = 324$

These equations form a system that can be solved using substitution or the quadratic formula.

We can express $y$ from the first equation: $y = 45 - x$ Substitute this into the second equation: $x(45 - x) = 324$ $45x - x^2 = 324$ Rearrange the equation: $x^2 - 45x + 324 = 0$

Step 4: Solve the quadratic equation

Now, we solve the quadratic equation $x^2 - 45x + 324 = 0$ using the quadratic formula: $x = \frac{-(-45) \pm \sqrt{(-45)^2 - 4(1)(324)}}{2(1)}$ $x = \frac{45 \pm \sqrt{2025 - 1296}}{2}$ $x = \frac{45 \pm \sqrt{729}}{2}$ $x = \frac{45 \pm 27}{2}$

So, the two possible values for $x$ are: $x = \frac{45 + 27}{2} = 36 \quad \text{or} \quad x = \frac{45 - 27}{2} = 9$

Step 5: Determine the correct solution

If $x = 36$, then from $x + y = 45$, we get $y = 45 - 36 = 9$.

If $x = 9$, then $y = 45 - 9 = 36$.

Thus, the father is 36 years old and the son is 9 years old.

Final Answer:

• The father is 36 years old.
• The son is 9 years old.

Would you like further clarification on this solution?

Here are some related questions for you:

1. How can you solve systems of equations involving quadratic expressions?
2. What is the quadratic formula and how is it derived?
3. How do you interpret word problems involving age-related relationships?
4. What would happen if the sum of their ages were different?
5. Can the product of their ages ever be negative?

Tip: Always double-check the signs when solving quadratic equations, especially in word problems!