If the sum of two numbers is 7 and sum of their cubes is 133, find the sum of their squares.

Let the two numbers be x and y.

Given,

Sum of two numbers is 7.

∴ x + y = 7

Sum of cubes of two numbers is 133.

∴ x³ + y³ = 133

By formula,

⇒ (x + y)³ = x³ + y³ + 3xy(x + y)

Substituting values we get :

⇒ 7³ = 133 + 3xy × 7

⇒ 343 = 133 + 21xy

⇒ 21xy = 343 - 133

⇒ 21xy = 210

⇒ xy = $\dfrac{210}{21}$ = 10.

By formula,

⇒ (x + y)² = x² + y² + 2xy

Substituting values we get :

⇒ 7² = x² + y² + 2 × 10

⇒ 49 = x² + y² + 20

⇒ x² + y² = 49 - 20 = 29.

Hence, sum of the squares of the numbers = 29.