How should i do this question??
it says:
find the equation of the tangent to the curve y=(x-2)^3 at the point (3,1). calculate the coordinates of the point where this tangent meets the curve again.
I know how to get the first part, if i'm not wrong, it's y = 3x - 8
but no matter how i do, i cant get another point that is meeting the curve again!
This is what i did for the second part:
y=3x-8 ---- (1)
y=(x-2)^3 ----(2)
(x-2)^3 - 3x + 8 = 0
simplify:
x^3 - 6x^2 + 9x = 0
x^2 - 6x + 9 = 0
factorize:
(x-3)^2=0
!?!??!
I'm pretty sure i did something wrong...
your tangent equation is correct.
also correct up to
x^3 - 6x^2 + 9x = 0
then
x(x^2 - 6x + 9) = 0
x(x-3)(x-3) = 0
so x = 0 or x = 3 (we knew that x=3 because we knew there was a tangent there)
You lost the x=0, a very common error
you can "lose" a constant if you factor it from a function f(x) = 0 but if you factor out a variable, that could be zero, thus a solution.
so when x=0, y = -8, so the other point of contact is (0,-8)
coordinates
To find the coordinates of the point where the tangent line meets the curve again, you need to solve the equation you obtained correctly.
Here's the correct solution:
You correctly found the equation of the tangent line as y = 3x - 8.
To find the coordinates of the point where this tangent line meets the curve again, we need to substitute the equation of the tangent line (y = 3x - 8) into the equation of the curve (y = (x - 2)^3).
Substituting y = 3x - 8 into y = (x - 2)^3, we get:
3x - 8 = (x - 2)^3
Now, let's solve this equation:
Expand the right side of the equation:
3x - 8 = x^3 - 6x^2 + 12x - 8
Rearrange the terms:
x^3 - 6x^2 + 9x = 0
Now, factor out an x:
x(x^2 - 6x + 9) = 0
Factor the quadratic expression:
x(x - 3)(x - 3) = 0
Now, we have three solutions:
x = 0, x = 3, x = 3
Since x = 0 is not a valid solution in this context (the point (0, 1) is not on the curve), we can discard it.
Therefore, the point where the tangent line meets the curve again is (3, 1).
So, the correct answer is the point (3, 1).
It seems you made a small mistake in your calculations where you factored x^2 - 6x + 9. The correct factorization is (x - 3)(x - 3).