I do not know how to find the coordinates of the curve where the tangent and the curve meet again.

Question

This is the question:

A curve has the equation $y= 3x^3 - 2x^2 + 2x$. The equation of the tangent to the curve when $x=1$ is $y = 7x + 4$. Find the coordinates of the curve where the tangent meets the curve again.

I tried to substitute the equation of the tangent into the equation of the curve but I got a cubic equation that did not give any possible solutions.

The answer to this question is $\left(-\dfrac 43, -\dfrac{40}3\right)$.

Answer 1

You are wrong about the tangent line. It is $y=7(x-1)+3=7x-4$. Using this, you will get the correct answer.

Answer 2

$y = 3x^3 - 2x^2 + 2x$

$x=1$ gives $y= 3$ and equation of tangent will be $y= 7x - 4$ in this case we have following system of equation to solve:

$y = 7x -4$

$y= 3x^3 -2 x^2 +2x$

subtracting these equations we get:

$3x^3-2x^2-5x+4=3x^3-2x^2-x-4(x-1)=0$

or : $(x-1)(3x^2+x-4)=0$

That gives $x=1$, $x=-4/3$