Let John's current age be $x$ years.

Step 1: Express Marc's age in terms of $x$

Since Marc is 3 times as old as John: $\text{Marc's age} = 3x$

Step 2: Two years ago

Two years ago:

• John's age was $x - 2$,
• Marc's age was $3x - 2$.

Step 3: Use the information about the sum of the squares

The sum of the squares of their ages two years ago was 178: $(x - 2)^2 + (3x - 2)^2 = 178$

Step 4: Expand the squares

Expand $(x - 2)^2$ and $(3x - 2)^2$: $(x - 2)^2 = x^2 - 4x + 4$ $(3x - 2)^2 = 9x^2 - 12x + 4$

Step 5: Combine terms

Add the two squares: $x^2 - 4x + 4 + 9x^2 - 12x + 4 = 178$ $x^2 + 9x^2 - 4x - 12x + 4 + 4 = 178$ $10x^2 - 16x + 8 = 178$

Step 6: Simplify the equation

Subtract 178 from both sides: $10x^2 - 16x + 8 - 178 = 0$ $10x^2 - 16x - 170 = 0$

Step 7: Divide through by 2

To simplify further, divide all terms by 2: $5x^2 - 8x - 85 = 0$

Final Answer:

The equation is: $5x^2 - 8x - 85 = 0$

Would you like me to solve the equation or explain further? Here are 5 related questions:

1. How do we solve $5x^2 - 8x - 85 = 0$?
2. What does this equation tell us about their ages?
3. How can the quadratic formula be applied to this problem?
4. Are there any real-life examples where similar age-related problems occur?
5. How can you verify the solution once the quadratic is solved?

Tip: Always check your expanded terms carefully to avoid errors in large problems.