Question

The sum of two numbers is 24 and the sum of their squares is 386,then one number is

Answer 1

$Given \: sum \: of \: two \: numbers \:is \:24.$

$Let \: one \: number = x$

$Second \: number = ( 24 - x )$

/* according to the problem given */

$\blue{ Sum \:of \:the \: squares = 386 }$

$\implies x^{2} + ( 24 - x )^{2} = 386$

$\implies x^{2} + 24^{2} - 2\times x \times 24 + x^{2} - 386 = 0$

$\implies 2x^{2} - 48x + 576 - 386 = 0$

$\implies 2x^{2} - 48x + 190 = 0$

/* Dividing each number by 2 , we get */

$\implies x^{2} - 24x + 190 = 0$

$\implies x^{2} -5x - 19x + 95 = 0$

$\implies x( x - 5) - 19( x - 5) = 0$

$\implies (x-5)(x-19) = 0$

$\implies x - 5 = 0 \: Or \: x - 19 = 0$

$\implies x = 5 \: Or \: x = 19$

Therefore.,

Case 1:

$If \: x = 5 \: then :$

$\red{ Required \:two \: numbers } \green {\: ( 5,19) }$

Case 2:

$If \: x = 19 \: then :$

$\red{ Required \:two \: numbers } \green {\: ( 19,5) }$

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