1. ## cubics

consider the equations of the cubic function and the straight line.

$\displaystyle y=x^3-x$, $\displaystyle y=ax+a$
The straight line crosses the cubic function at point A(-1,0). Suppose that the straight line crosses the cubic function at two further points P($\displaystyle x_1,y_1$) and Q$\displaystyle (x_2,y_2$), where $\displaystyle x_1<x_2$.
1. Show that $\displaystyle x_1+x_2=1$
2. Hence find the equation of the line throught the point A tangent to the cubic function at a point distinct from the point A

2. ## Cubic

Hello requal
Originally Posted by requal
consider the equations of the cubic function and the straight line.

$\displaystyle y=x^3-x$, $\displaystyle y=ax+a$
The straight line crosses the cubic function at point A(-1,0). Suppose that the straight line crosses the cubic function at two further points P($\displaystyle x_1,y_1$) and Q$\displaystyle (x_2,y_2$), where $\displaystyle x_1<x_2$.
1. Show that $\displaystyle x_1+x_2=1$
2. Hence find the equation of the line throught the point A tangent to the cubic function at a point distinct from the point A
$\displaystyle y = ax + a$ meets $\displaystyle y = x^3 -x$ where

$\displaystyle x^3 - x(a+1) - a = 0$

$\displaystyle x = -1$ is a root of this equation, so $\displaystyle (x+1)$ is a factor.

Factorise:

$\displaystyle (x+1)(x^2 - x - a) = 0$

Then consider the roots of the quadratic. For question (1), say that their sum = $\displaystyle -\frac{b}{a}$. For question (2), say that they must be equal if the line is tangent; so $\displaystyle b^2 = 4ac$.

Can you do it now?

3. Slight warning: when Grandad says " For question (1), say that their sum = $\displaystyle -\frac{b}{a}$" and "For question (2), say that they must be equal if the line is tangent; so $\displaystyle b^2= 4ac$", he is referring to the "a, b, c" in the general quadratic, $\displaystyle ax^2+ bx+ c$, not the "a" in this particular equation, y= ax+ a.