Welcome to Westonci.ca, where finding answers to your questions is made simple by our community of experts. Join our Q&A platform and get accurate answers to all your questions from professionals across multiple disciplines. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
Answer:
3
Step-by-step explanation:
since the 3 integers are consecutive, we are dealing with x, x+1, x+2.
and their sum is the same as their product :
x + (x + 1) + (x + 2) = x(x + 1)(x + 2)
3x + 3 = x(x² + 3x + 2) = x³ + 3x² + 2x
x³ + 3x² - x - 3 = 0
this is a polynomial of third degree.
and as such it has 3 solutions.
of course, it could be that some of them are the same or are even in the realm of complex numbers (i = sqrt(-1)), but usually these 3 solutions are different real numbers.
I tried x=1 just to see, and, hey, it is a solution for this equation.
x = 1 means that the other 2 consecutive integers are 2 and 3.
and indeed, 1+2+3 = 1×2×3 = 6.
now it is easier to find the other 2 solutions, as a zero solution can be expressed as a factor of the whole expression.
for x = 1 the factor term is (x - 1), as this term is then turning 0, when x = 1.
I can divide the main expression by this factor and then analyze the quotient about the other 2 solutions.
x³ + 3x² - x - 3 : x - 1 = x² + 4x + 3
- x³ - x²
----------------
0 4x² - x
- 4x² - 4x
-----------------------
0 + 3x - 3
- 3x - 3
---------------------------
0 0
so, the original expression can be written as
(x² + 4x + 3)(x - 1).
now we need to find the 2 zero solutions for x²+4x+3
the general solution to a quadratic equation is
x = (-b ± sqrt(b² - 4ac))/(2a)
in our case
a = 1
b = 4
c = 3
so,
x = (-4 ± sqrt(4² - 4×1×3))/(2×1) =
= (-4 ± sqrt(16 - 12))/2 = (-4 ± sqrt(4))/2 =
= (-4 ± 2)/2 = -2 ± 1
x1 = -2 + 1 = -1
x2 = -2 - 1 = -3
so, we have the additional solutions :
-1 0 1
-3 -2 -1
-1 + 0 + 1 = -1×0×1 = 0
-3 + -2 + -1 = -3×-2×-1 = -6
and there we have it fully proven :
there are 3 different sets of 3 consecutive integers with the same sum as product.
Thanks for using our platform. We're always here to provide accurate and up-to-date answers to all your queries. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Westonci.ca is your trusted source for answers. Visit us again to find more information on diverse topics.