The product of four consecutive positive integers is 1 less than $461^2$. What is the least of these four numbers?

The least of the four numbers is 20

Step-by-step explanation:

The given information are;

The product of 4 consecutive numbers = 461² - 1

The required information is the least among the four numbers

Let the numbers be N-2, N-1, N, N + 1

We have;

(N - 2)×(N)×(N + 1)(N - 1) = 461² - 1

(N - 2)×(N)×(N + 1)(N - 1) - (461² - 1) = 0

Which gives;

N⁴ - 2·N³ - N² + 2·N - 212520 = 0

Factorizing online also gives using ;

(N + 21)(N - 22)(N² - N + 460) = 0

Therefore;

N = 22 or -21 are possible solutions

However, the requirement for positive integers give the possible solution as N = 22

Therefore, where the four numbers are N-2, N-1, N, N + 1, we then have;

22 -2, 22 -1, 22, 22 + 1 which gives the numbers as 20, 21. 22, and 23

The least of the four numbers is therefore 20.

step-by-step explanation:

equation

you are asked to write the equation

  expression = 0

using the given expression. when you do that, you have

  2x^2-2x-12 = 0

factoring

you can see that each of the coefficients is even, so a factor of 2 can be removed immediately.

  2(x^2 -x -6) = 0

if you multiply the binomials (x+a)(x+b), you get

  (x +a)(x +b) = x^2 +(a+b)x +ab

this tells you the coefficient of x is the sum of two factors of the constant term. here, that means -1 is the sum of two factors of -6. we want to know those factors so we can factor the trinomial.

  -6 = -1×6 = -2×3 = -3×2 = -6×1

the factor pair that sums to -1 is {-3, 2}, so our factorization is

  2(x -3)(x +2) = 0

solutions

the solutions to this equation are the values of x that make the product zero. a product is zero only when only of the factors of the product is zero. here, that means

  x - 3 = 0

  x = 3 . . . add 3 to both sides of the equation

or

  x +2 = 0

  x = -2 . . . add -2 to both sides of the equation

the solutions to the equation are x=3 or x=-2. both of these values of x satisfy the equation—they make it be true.

i think it will be zero as slope has a formula of difference between initial and final distance divided by itz time interval