The sum of the squares of two numbers is 233 and one of the numbers is

1.	The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers.
Answer» Let one of the number be x. Then the other number will be 2x – 3. From the question: x²+ (2x – 3)² = 233 ⇒ x²+ 4x²+ 9 – 12x = 233 ⇒ 5x²– 12x – 224 = 0 ⇒ 5x²– 40x + 28x – 224 = 0 ⇒ 5x(x – 8)+ 28(x – 8) = 0 ⇒ (5x+ 28) (x – 8) = 0 Now, 5x + 28 cannot be 0 so, x – 8 = 0 ⇒ x = 8 Considering the value of x = 8, we have 2x – 3 = 15 Thus, the two numbers are 8 and 15 respectively.

The sum of the squares of two numbers is 233 and one of the numbers is 3 less than twice the other number. Find the numbers.

Answer»

Let one of the number be x. Then the other number will be 2x – 3.

From the question:

x²+ (2x – 3)² = 233

⇒ x²+ 4x²+ 9 – 12x = 233

⇒ 5x²– 12x – 224 = 0

⇒ 5x²– 40x + 28x – 224 = 0

⇒ 5x(x – 8)+ 28(x – 8) = 0

⇒ (5x+ 28) (x – 8) = 0

Now, 5x + 28 cannot be 0

so, x – 8 = 0 ⇒ x = 8

Considering the value of x = 8, we have

2x – 3 = 15

Thus, the two numbers are 8 and 15 respectively.