Question

The difference of squares of two numbers is 180. If the square of the smaller number is 8 times the larger number, find both numbers.

Hint:

Word problems → form equations first. Steps:
Assign variables clearly.
Convert statements into equations.
Substitute to reduce variables.
Always check which root is valid.

Answer 1

Concept: Translate the given statements into equations and solve simultaneously. Let: \[ \text{Smaller number} = x, \quad \text{Larger number} = y \] Step 1: Form equations Difference of squares: \[ y^2 - x^2 = 180 \quad \cdots (1) \] Square of smaller number is 8 times the larger: \[ x^2 = 8y \quad \cdots (2) \] Step 2: Substitute equation (2) into (1) \[ y^2 - 8y = 180 \] \[ y^2 - 8y - 180 = 0 \] Step 3: Solve quadratic \[ y^2 - 8y - 180 = 0 \] Factorization: \[ (y - 18)(y + 10) = 0 \] \[ y = 18 \quad \text{or} \quad y = -10 \] Since larger number must be positive: \[ y = 18 \] Step 4: Find $x$ Using $x^2 = 8y$: \[ x^2 = 8 \times 18 = 144 \] \[ x = 12 \quad (\text{taking positive value}) \] Final Answer: \[ \boxed{\text{Smaller number } = 12, \quad \text{ Larger number } = 18} \]