If 𝐴 has coordinates negative seven, 𝑥 and 𝐵 has coordinates nine, 14, where 𝐴𝐵 equals four root 17 length units, find all possible values of 𝑥.
Let’s begin by thinking about this geometrically. Here is point 𝐵 with coordinates nine, 14. It lies in the first quadrant. We know point 𝐴 has an 𝑥-coordinate of negative seven. This means that point 𝐴 must lie somewhere on the line 𝑥 equals negative seven. For point 𝐴 to be a distance of four root 17 away from 𝐵, it could lie in two possible places. One is likely to be a little bit above point 𝐵 in the 𝑦-direction, and one is likely to be below it. Now, at this stage, we don’t necessarily know whether one of these points lies below the 𝑥-axis. We’re just making a guess. But we do know we’re looking for two possible values of 𝑥. So, how do we find these possible values?
Imagine we have two points on the coordinate plane with coordinates 𝑥 one, 𝑦 one and 𝑥 two, 𝑦 two. The distance between them comes from the Pythagorean theorem. It’s 𝑑 is equal to the square root of 𝑥 two minus 𝑥 one squared plus 𝑦 two minus 𝑦 one squared. And it doesn’t matter which way round we decide to choose our coordinates. Let’s let 𝑥 one, 𝑦 one be negative seven, 𝑥 and 𝑥 two, 𝑦 two be nine, 14. Then, the distance between these two points is the square root of nine minus negative seven squared plus 14 minus 𝑥 squared. In fact, we can simplify a little bit by recognizing that nine minus negative seven is 16. So we get 256, that’s 16 squared, plus 14 minus 𝑥 squared inside our root. Then, we also note that the length between 𝐴 and 𝐵 is given as four root 17 length units. So this entire expression must be equal to four root 17.
We need to solve this equation for 𝑥. So let’s begin by squaring both sides. When we do, four root 17 squared is 16 times 17, which is 272. And on the right-hand side, we simply have 256 plus 14 minus 𝑥 squared. Then, we have two techniques that we can use to solve this remaining equation. We could set it equal to zero and solve as any other quadratic equation. Alternatively, let’s see what happens if we subtract 256 from both sides. If we subtract 256, our equation becomes 16 equals 14 minus 𝑥 squared. To solve for 𝑥, we’re going to need to take the square root of both sides. But this means we have to take both the positive and negative square root of 16. So, we get plus or minus root 16 is equal to 14 minus 𝑥. And of course root 16 is four. So, we find that positive or negative four is equal to 14 minus 𝑥.
We now need to split this into two separate equations. The first will be four equals 14 minus 𝑥, and the second will be negative four equals 14 minus 𝑥. From both of these equations, we’ll subtract 14 from each side. That gives us negative 10 equals negative 𝑥 and negative 18 equals negative 𝑥, respectively. Finally, we multiply through by negative one, and this will give us the values of 𝑥. That’s 𝑥 equals 10 and 𝑥 equals 18. And so we have our two possible values of 𝑥. They are 𝑥 equals 10 or 𝑥 equals 18.