Question

addition of two numbers is 34 and addition of square of both numbers is 650. Find the smallest number​

Answer 1

Step-by-step explanation:

let the two numbers be x and y

x + y = 34

y = 34 - x

x² + y² = 650

substitute y = 34 - x in above equation

${x}^{2} + {(34 - x)}^{2} = 650 \\ {x}^{2} + ( {34}^{2} + {x}^{2} - 68x) = 650\\ 2 {x}^{2} - 68x + 1156 = 650 \\ 2 {x}^{2} - 68x + 1156 - 650 = 0 \\ 2 {x}^{2} - 68x + 506 = 0 \\ 2( {x}^{2} - 34x + 253) = 0 \\ {x}^{2} - 34x + 253 = 0 \\ {x}^{2} - 11x - 23x + 253 = 0 \\ x(x - 11) - 23(x - 11) = 0 \\ (x - 11)(x - 23) = 0 \\ x - 11 = 0 \: \: \: \: \: \: x - 23 = 0 \\ x = 11 \: \: \: \: \: \: x = 23$

x = 11 or x = 23

y = 34 - 11 = 23

or

y = 34 - 23 = 11

therefore the smallest number of the two is 11

Hope you get your answer

Answer 2

Given:

a + b = 34........(i)

a² + b² = 650.........(ii)

Now, According to Question

➟ (a + b)² = (34)²

➟ a² + b² + 2ab = 1156

Use the value of eq(ii)

➟ a² + b² + 2ab = 1156

➟ 650 + 2ab = 1156

➟ 2ab = 1156 - 650

➟ 2ab = 506---------(iii)

Now, subtract (iii) from (ii)

➟ (a² + b²) - (2ab) = (650) - (506)

➟ a² + b² - 2ab = 650 - 506

➟ (a - b)² = 144

On removing square

➟ a - b = √144

➟ a - b = 12-------------(iv)

On solving (i) & ((iv)

➟ (a + b) - (a - b) = 34 - 12

➟ a + b - a + b = 22

➟ 2b = 22

➟ b = 22/2

➟ b = 11

And

➟ a - b = 12

➟ a - 11 = 12

➟ a = 12 + 11

➟ a = 23

Therefore:-

The smallest number is 11.