Question

One positive number exceeds another by 5. The sum of their squares is 193. Find both numbers.​ explain how did you find the two nunbers​

Answer 1

Step-by-step explanation:

x^2 + (x+5)^2 = 193

FOIL (x+5)(x+5)

x^2 + x^2 + 10x + 25 = 193

2x^2 + 10x + 25 - 193 = 0

2x^2 + 10x - 168 = 0

simplify, divide by 2

x^2 + 5x - 84 = 0

easily factors to

(x+12)(x-7) = 0

the positive solution

x = 7

:

:

Check: 7^2 + 12^2 =

49 + 144 = 193

Answer by Theo(10715) (Show Source): You can put this solution on YOUR website!

x is the value of one of the numbers.

y is the value of the other number.

x = y + 5 is one of your equations.

x^2 + y^2 = 193 is the other of your equations.

in the second equation, replace x with y + 5 to get:

(y + 5)^2 + y^2 = 193

simplify to get:

y^2 + 10y + 25 + y^2 = 193

combine like terms to get:

2y^2 + 10y + 25 = 193

subtract 193 from both sides of the equation to get:

2y^2 + 10y - 168 = 0

divide both sides of this equation by 2 to get:

y^2 + 5y - 84 = 0

factor to get:

(y + 12) * (y - 7) = 0

solve for y to get:

y = -12 or y = 7

value of y has to be positive, so y = 7 is a possible solution.

x = y + 5, therefore x = 12

your number appear to be 12 and 7.

12 - 7 = 5, which satisfies one of the requirements.

12^2 + 7^2 = 144 + 49 = 193, which satisfies the other requirement.

your solution is that the numbers are 7 and 12.

Answer 2

Let x and (x + 5) be the given numbers.
Now, according to the question

$x^{2} + (x + 5)^{2} = 193$                                      .................... (Given)
$x^{2} + x^{2} + 10x + 25 = 193$           ......... (using identity $(x + y)^{2} = x^{2} + y^{2} + 2xy$)
$2x^{2} + 10x +25 - 193 = 0$
$2x^{2} + 10x - 168 = 0$
Taking 2 common from all the terms,
$2(x^{2} + 5x - 84) = 0$
$x^{2} + 5x - 84 = \frac{0}{2}$
$x^{2} + 5x - 84 = 0$
Now using the factorization method,
$x^{2} + 12x - 7x - 84 = 0$
$x (x + 12) - 7 (x + 12) = 0$
$(x + 12) (x - 7) = 0$
Now equating both factors with 0,
$(x + 12) = 0\\x = -12$           (This is not valid as given positive integer so we cannot
                               take negative values.)
$(x - 7) = 0\\x = 7$

∴ Value of our first positive number = x= 7.

Value of second positive number = x + 5
                                                       = 7 + 5       ⇒ 12

∴ Value of second positive number = x + 5 = 12.

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