Find all the following value for x for which distance between point A(x,-1) and B(5,3) is 5units

x = 2, x = 8

Step-by-step explanation:

Calculate the distance using the distance formula and equate to 5

d = √ (x₂ - x₁ )² + (y₂ - y₁ )²

with (x₁, y₁ ) = A (x, - 1) and (x₂, y₂ ) = B(5, 3)

d =

   =

   = , thus

= 5 ( square both sides )

(5 - x)² + 16 = 25 ( subtract 16 from both sides )

(5 - x)² = 9 ( take the square root of both sides )

5 - x = ± = ± 3 ( subtract 5 from both sides )

- x = - 5 ± 3 , thus

- x = - 5 + 3 = - 2 ( multiply both sides by - 1 )

x = 2

or

x = - 5 - 3 = - 8 ( multiply both sides by - 1 )

x = 8

Step-by-step explanation:

The formula to find the distance between 2 points in the coordinate plane is

We have our distance, and we also have all the coordinates but the first x.  Fillling in with what we have gives us this:

which simplifies to

. Expanding that binomial gives us:

.  Combining like terms gives us:

which is the same thing as above, only in standard form for polynomials. Now we need to get that x out from under that square root sign. We do that by squaring both sides to get:

. Now we have to factor to solve for x. We'll put everything on one side of the equals sign, set the polynomial equal to 0, then factor.

is our polynomial now. a = 1, b = -10, c = 16. The product ac is 1 * 16 which is 16. Some combination of the factors of 16 will result in a -10. So we need the factors of 16.

16: {1, 16}, {2, 8}, {4, 4}

The only combination of those factors that will result in a -10 is the second pair, {2, 8}. If we add 2 and 8 we get 10, but in order for our 10 to be negative, both 2 and 8 have to be negative. So we rewrite the polynomial in terms of -2 and -8:

Now we can factor by grouping. Group the first 2 terms together and the second 2 terms together without moving any of their positions:

From each set of parenthesis we will now factor out what's common.  x is common in the first set of ( ), and 2 is common in the second set of ( ):

What's common now is the binomial (x - 8). So we'll factor that out now:

By the Zero Product Property, either

x - 8 = 0 or x - 2 = 0.

If x - 8 = 0, then x = 8. If x - 2 = 0, then x = 2.

It looks like we have 2 solutions. Let's try them both and see if, when we stick an 8 and then a 2 into our distance formula, the distance is 5:

is

is

is

which does in fact equal 5. Now let's try the 2:

which is

is

is

which also comes out to equal 5.

So the 2 values of x which will work here are 2 and 8.

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step-by-step explanation:

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step-by-step explanation: