Question
The age of a father is twice the square of his son's age. Ten years from now, the father's age will be four times his son's age. Find their present ages.
Solution — Step by Step
Word problems with quadratics have two parts: setting up the equations (the hard part) and solving (the mechanical part). Most marks are allocated to correct setup.
Step 1: Assign variables.
Let the son's present age = x years.
Then, the father's present age = 2x² years (given: "twice the square of the son's age").
Step 2: Use the second condition to form an equation.
"Ten years from now":
- Son's age = x + 10
- Father's age = 2x² + 10
"Father's age will be four times the son's age":
2x² + 10 = 4(x + 10)
Step 3: Expand and simplify.
2x² + 10 = 4x + 40 2x² − 4x + 10 − 40 = 0 2x² − 4x − 30 = 0
Divide throughout by 2:
x² − 2x − 15 = 0
Step 4: Solve the quadratic equation.
We need two numbers that multiply to −15 and add to −2. Those are −5 and 3.
x² − 5x + 3x − 15 = 0 x(x − 5) + 3(x − 5) = 0 (x + 3)(x − 5) = 0
x = −3 or x = 5
Step 5: Reject the invalid root.
Age cannot be negative. So x = −3 is rejected.
Son's age = 5 years. Father's age = 2(5)² = 50 years.
Verification:
- Present: Father = 50 = 2 × 25 = 2 × (5)² ✓
- Ten years later: Son = 15, Father = 60. Is 60 = 4 × 15? Yes ✓
Answer
Son's present age: 5 years
Father's present age: 50 years
Why This Works
The key insight is translating each English sentence into a mathematical expression, one piece at a time. "Twice the square" = 2x². "Ten years from now" adds 10 to each current age. "Four times" means multiply by 4. Break down the language systematically.
Once the equation is set up, the rest is standard quadratic solving. The critical final step — rejecting negative roots because age can't be negative — is where many students forget to apply real-world context.
Alternative Method: Direct Verification
Once you have the quadratic x² − 2x − 15 = 0, you can also use the quadratic formula:
D = 4 + 60 = 64, √D = 8 x = (2 ± 8) / 2
x = 5 or x = −3. Same answer.
💡 Expert Tip
In word problems, always ask at the end: "Does this answer make sense in the real world?" Age, distance, and length cannot be negative. Number of items must be a positive integer. Area must be positive. These common sense checks are how you know which root to keep.
Common Mistake
⚠️ Common Mistake
Mistake: Writing both roots as the answer without checking.
If you write "x = 5 or x = −3, so ages are 5 and 50 OR −3 and 18," you'll lose marks. The question asks for a unique answer, and context tells us age must be positive. Always state explicitly: "Since age cannot be negative, x = −3 is rejected."
In CBSE board exams, this rejection step with justification is worth 1 mark.