Find all points on the x-axis that are 10 units from the point (7, 6). (Hint: First write the distance formula with (7, 6) as one of the points and (x, 0) as the other point.)
To find all the points on the x-axis that are 10 units from the point (7, 6), we can use the distance formula. Let's denote the x-coordinate of the point on the x-axis as x.
The distance formula is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, (x1, y1) = (7, 6) and (x2, y2) = (x, 0).
Substituting the values into the distance formula, we have:
10 = sqrt((x - 7)^2 + (0 - 6)^2)
Simplifying:
100 = (x - 7)^2 + 36
Expand the equation:
100 = x^2 - 14x + 49 + 36
Combine like terms:
x^2 - 14x - 87 = 0
To solve for x, we can either factor the quadratic equation or use the quadratic formula. Let's use the quadratic formula:
x = (-(-14) ± sqrt((-14)^2 - 4(1)(-87))) / (2(1))
Simplifying:
x = (14 ± sqrt(196 + 348))/2
x = (14 ± sqrt(544)) / 2
x = (14 ± 2sqrt(136)) / 2
Simplifying further:
x = 7 ± sqrt(136)
Therefore, there are two points on the x-axis that are 10 units from the point (7, 6). One point is (7 + sqrt(136), 0) and the other point is (7 - sqrt(136), 0).
To find all points on the x-axis that are 10 units from the point (7, 6), we can use the distance formula.
The distance formula is given by:
d = sqrt((x2 - x1)^2 + (y2 - y1)^2)
In this case, we want to find the points that are 10 units away from the point (7, 6). Let's assume that one of the points is (7, 6), and the other point is (x, 0), since we are interested in points on the x-axis.
Using the distance formula, we can substitute the values into the formula:
10 = sqrt((x - 7)^2 + (0 - 6)^2)
Simplifying:
100 = (x - 7)^2 + 6^2
100 = (x - 7)^2 + 36
100 - 36 = (x - 7)^2
64 = (x - 7)^2
To solve for x, we take the square root of both sides:
sqrt(64) = sqrt((x - 7)^2)
8 = abs(x - 7)
This gives us two possibilities:
1) x - 7 = 8
x = 8 + 7
x = 15
2) x - 7 = -8
x = -8 + 7
x = -1
Therefore, the two points on the x-axis that are 10 units from the point (7, 6) are (15, 0) and (-1, 0).
√( (x-7)^2 + (0-6)^2 ) = 10
square both sides and expand
x^2 - 14x + 49 + 36 = 100
x^2 - 14x - 15 = 0
(x - 15)(x + 1) = 0
x = 15 or x = -1
interpret my results.