cubics

**requal** · January 8th 2009, 09:36 PM

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

$y=x^3-x$ , $y=ax+a$
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( $x_1,y_1$ ) and Q $(x_2,y_2$ ), where $x_1<x_2$ .
1. Show that $x_1+x_2=1$
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

**Grandad** · January 9th 2009, 01:13 AM

Hello requal

Originally Posted by requal

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

$y=x^3-x$ , $y=ax+a$
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( $x_1,y_1$ ) and Q $(x_2,y_2$ ), where $x_1<x_2$ .
1. Show that $x_1+x_2=1$
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

$y = ax + a$ meets $y = x^3 -x$ where

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

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

Factorise:

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

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

Can you do it now?

Grandad

**HallsofIvy** · January 9th 2009, 04:45 AM

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

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