Question

find two consecutive positive integers, sum of whose squares is 613

Answer 1

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Related Questions

Find two consecutive positive integers, sum of whose square is 613.

A. 17 and 18

B. 20 and 22

C. 16 and 17

D. 15 and 18

Answer

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Hint: Consider x and x+1 be two consecutive numbers and they given sum of squares of these consecutive numbers are 613 i.e x2+(x+1)2 =613. Hence solve this quadratic equation and determine the value of x

Complete step-by-step answer:

Let the two consecutive positive integers be ‘x’ and ‘x + 1’

According to the question,

x2 + (x + 1)2 = 613x2 + x2 + 12+ 2x = 6132x2 + 2x = 613 12x2+ 2x = 6122x2 + 2x − 612 = 02 (x2 + x − 306) = 0x2 + x − 306 = 0

We have got a quadratic equation now,

x² + x - 306 = 0

x² - 18x + 17x - 306 = 0

x (x - 18) + 17 (x - 18) = 0

(x - 18) (x + 17) = 0

x = 17 or x = - 18

But x is given to be a positive integer. Therefore, x ≠ -18, x = 17.

Thus, the two consecutive positive integers are 17 and 18.

Note:-Numbers that follow each other continuously in the order from smallest to largest are called consecutive numbers For Ex. 1,2,3,4,5 .Students should know the basic expansion formula of (a+b)2=a2+b2+2ab to solve these types of problems. if I have answer correctly please mark me as a brainlist answers

Answer 2

$\huge \purple{ \underline{ \boxed{ \red{ \mathbb{ANSWER : }}}}}$

$\red{ \textsf{The two consecutive positive integers sum of whose squares is 613 are\orange{ 17 and 18 .}}}$

$\huge \purple{ \underline{ \boxed{ \red{ \mathbb{EXPLANATION : }}}}}$

$\large\green{ \underline{ \underline{ \mathbb{GIVEN : }}}}$

$\orange{ \textsf{Sum of squares of two positive integers is 613 .}}$

$\large\green{ \underline{ \underline{ \mathbb{FORMULA \: USED : }}}}$

$\orange{ \textsf{( a + b ) ² = a² + b² + 2ab}}$

$\large\green{ \underline{ \underline{ \mathbb{SOLUTION : }}}}$

$\red{ \textsf{Let x be the first number and ( x + 1 ) be the second number . }}$

$\red{ \textsf{Therefore}}$

$\red{ \longmapsto \mathsf{ {x}^{2} + {(x + 1)}^{2} = 613}}$

$\red{ \longmapsto \mathsf{ {x}^{2} + {x}^{2} + 1 + 2x = 613}}$

$\red{ \longmapsto \mathsf{ {2x}^{2} + 2x + 1 = 613}}$

$\red{ \longmapsto \mathsf{ {2x}^{2} + 2x = 613 - 1}}$

$\red{ \longmapsto \mathsf{ {2x}^{2} + 2x = 612}}$

$\red{ \longmapsto \mathsf{ {2x}^{2} + 2x - 612 = 0}}$

$\red{ \longmapsto \mathsf{ 2({x}^{2} + x - 306) = 0}}$

$\red{ \longmapsto \mathsf{ {x}^{2} + x - 306 = 0}}$

$\red{ \longmapsto \mathsf{ {x}^{2} +18 x - 17x - 306 = 0}}$

$\red{ \longmapsto \mathsf{ x( x+18 ) - 17(x + 18) = 0}}$

$\red{ \longmapsto \mathsf{ ( x - 17)( x+18 )= 0}}$

$\red{ \begin{array}{l | l} \longmapsto \mathsf{ x - 17 = 0 }& \mathsf{x+18= 0} \\ \longmapsto \mathsf{ x = 17 }& \mathsf{x = - 18} \end{array}}$

$\red{ \textsf{Therefore ,}}$

$\red{ \textsf{First number = 17 or -18}}$

$\red{ \textsf{Second number = 18 or -17}}$

$\large \green{ \underline{ \underline{ \mathbb{KNOW \: MORE : }}}}$

$\orange{\mathbb{INTEGERS : }}$

An integer is a whole number not a fractional number that can be positive , negative , or zero .

$\orange{ \mathbb{SQUARE \: NUMBER :}}$

A square number or perfect square is an integer that is the square of an integer ; in other words , it is the product of some integer with itself .