Question

The sum of two numbers is 20 and the sum of their squares is 232. Find the numbers.

Answer 1

Answer:

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Answer 2

The numbers are 6 and 14

Given :

The sum of two numbers is 20 and the sum of their squares is 232

To find :

The numbers

Solution :

Step 1 of 2 :

Form the equation to find the numbers

Let the numbers are x and y

Since sum of two numbers is 20

∴ x + y = 20 - - - - - (1)

Again sum of their squares is 232

∴ x² + y² = 232 - - - - - (2)

Step 2 of 2 :

Find the numbers

From Equation 1 we get

y = 20 - x - - - - - (3)

Putting y = 20 - x in Equation 2 we get

$\displaystyle \sf {x}^{2} + {(20 - x)}^{2} = 232$

$\displaystyle \sf{ \implies } {x}^{2} + 400 - 40x + {x}^{2} = 232$

$\displaystyle \sf{ \implies } 2{x}^{2} - 40x + 168 = 0$

$\displaystyle \sf{ \implies } {x}^{2} - 20x + 84 = 0$

$\displaystyle \sf{ \implies } {x}^{2} - (14 + 6)x + 84 = 0$

$\displaystyle \sf{ \implies } {x}^{2} - 14 x - 6x+ 84 = 0$

$\displaystyle \sf{ \implies } x(x - 14) - 6(x - 14) = 0$

$\displaystyle \sf{ \implies } (x - 14) (x - 6) = 0$

x - 14 = 0 gives x = 14

x - 6 = 0 gives x = 6

When x = 14

From Equation 3 we get

y = 20 - 14 = 6

When x = 6

From Equation 3 we get

y = 20 - 6 = 14

Hence the required numbers are 6 and 14

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