Question

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

Hint:

When a problem involves squares and differences, express one variable in terms of the other and reduce it to a quadratic equation.

Answer 1

Step 1: Let the two numbers be \( x \) and \( y \).
Let \( x \) be the larger number and \( y \) be the smaller number. Given: \[ x^2 - y^2 = 180 \quad \text{(1)} \] and \[ y^2 = 8x \quad \text{(2)} \]
Step 2: Substitute equation (2) in equation (1).
\[ x^2 - 8x = 180 \] \[ x^2 - 8x - 180 = 0 \]
Step 3: Solve the quadratic equation.
We can factorize it as: \[ x^2 - 18x + 10x - 180 = 0 \] \[ x(x - 18) + 10(x - 18) = 0 \] \[ (x - 18)(x + 10) = 0 \] \[ x = 18 \text{ or } x = -10 \] Since \( x \) represents a number whose square is positive, take \( x = 18 \).
Step 4: Substitute in (2) to find \( y \).
\[ y^2 = 8x = 8(18) = 144 \] \[ y = 12 \] Step 5: Final Answer.
\[ \boxed{x = 18, \, y = 12} \]