Find x, (Enter your answers as a comma-separated list.)

Question

Find x so that the distance between (−2, 9) and (x, 7) is 29

  

Answer 1

We will set this up as an equation based on the distance formula, d = √(x₂ - x₁)² + ( y₁ - y₂)². Plugging in the values we are given, along with the actual distance, the equation becomes √29 = √( x + 2 )² + ( 7 - 9 )². Working under the radical to FOIL the first set of ( ) and simplify the second set of ( ), we have, under the radical, x² + 4x + 4 + 4. Combining what we can, the full equation is now √29 = √x² + 4x + 8. Squaring both sides of the equation yields 29 = x² + 4x + 8.

Since the right side is not a perfect square trinomial, we will turn it into one so that we can finish solving for x by taking square root of both sides. Moving the 8 over, we have 21 = x² + 4x. Now we complete the square:

21 + 4 = x² + 4x + 4 → 25 = (x + 2 )² → ± √25 = x + 2 → ±5 = x + 2, so x = 3 or x = -7.

I'm not sure what the last part of the problem means when it says to "write your answers as a comma separated list" except to write the x-values as 3, -7. Checking both of these values via the distance formula confirms they are the correct x-coordinates for the two points that have the given distance from the point ( -2, 9 ).