Tangent to a Curve

[TangoFoxtrotGolf]

Find a point P on the curve y = x[sup:1o6xq0tm]3[/sup:1o6xq0tm] such that the slope of the line passing through P and (1, 1) is 3/4.

p[sub:1o6xq0tm]1[/sub:1o6xq0tm] = (1, 1)

p[sub:1o6xq0tm]2[/sub:1o6xq0tm] = (x, x[sup:1o6xq0tm]3[/sup:1o6xq0tm])

m = (y[sub:1o6xq0tm]2[/sub:1o6xq0tm] - y[sub:1o6xq0tm]1[/sub:1o6xq0tm])/(x[sub:1o6xq0tm]2[/sub:1o6xq0tm] - x[sub:1o6xq0tm]1[/sub:1o6xq0tm])

3/4 = (x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 1)/(x - 1)

3/4 - 3/4 = (x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 1)/(x - 1) - 3/4

(x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 1)/(x - 1) - 3/4 = 0

4(x - 1)[(x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 1)/(x - 1)] - (3/4)(4)(x - 1) = 0

4(x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 1) - 3(x - 1) = 0

(4x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 4) - (3x - 3) = 0

4x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 4 - 3x + 3 = 0

4x[sup:1o6xq0tm]3[/sup:1o6xq0tm] - 3x - 1 = 0

I don't know where to go from here?

Did I go wrong somewhere up above?

 

[Denis]

p[sub:21lyal3a]1[/sub:21lyal3a] = (1, 1)
p[sub:21lyal3a]2[/sub:21lyal3a] = (x, x[sup:21lyal3a]3[/sup:21lyal3a])
m = (y[sub:21lyal3a]2[/sub:21lyal3a] - y[sub:21lyal3a]1[/sub:21lyal3a])/(x[sub:21lyal3a]2[/sub:21lyal3a] - x[sub:21lyal3a]1[/sub:21lyal3a])
3/4 = (x[sup:21lyal3a]3[/sup:21lyal3a] - 1)/(x - 1)
3/4 - 3/4 = (x[sup:21lyal3a]3[/sup:21lyal3a] - 1)/(x - 1) - 3/4
(x[sup:21lyal3a]3[/sup:21lyal3a] - 1)/(x - 1) - 3/4 = 0
4(x - 1)[(x[sup:21lyal3a]3[/sup:21lyal3a] - 1)/(x - 1)] - (3/4)(4)(x - 1) = 0
4(x[sup:21lyal3a]3[/sup:21lyal3a] - 1) - 3(x - 1) = 0
(4x[sup:21lyal3a]3[/sup:21lyal3a] - 4) - (3x - 3) = 0
4x[sup:21lyal3a]3[/sup:21lyal3a] - 4 - 3x + 3 = 0
4x[sup:21lyal3a]3[/sup:21lyal3a] - 3x - 1 = 0

Correct; but you tired me out with all those steps; can be done much simpler/quicker by getting the
y-intercept, which is 1/4, hence y = (3/4)x + 1/4 ....

4x^3 - 3x - 1 = 0
Solvable by inspection:
(2x + 1)(2x + 1)(x - 1) ; see that?