The graphs of $y = \frac{x^3}{2} - 4x - 1$ and $y = 1 - x$ are drawn on the grid.

(a)
The points of intersection of the curve and the straight line give the solutions of a cubic equation. Find the cubic equation, giving your answer in the form $ax^3+bx^2+cx+d=0$ , where a, b, c and d are integers.

Answer ...
(b)
By drawing a tangent, find the gradient of the curve at $(-2, 3$ ).

Answer ...
(c)
The equation $x^3-4x=0$ can be solved by drawing a suitable straight line on the grid.
1. (i)
  Find the equation of the straight line.
  
  Answer ...
2. (ii)
  By drawing this straight line, solve the equation $x^3-4x=0$ .
  
  Answer x = ... or ... or ...

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Worked solution

1. Equate the curve and straight line, then rearrange to zero.

At the points of intersection,
$\frac{x^3}{2}-4x-1=1-x.$
Bring all terms to one side:
$\frac{x^3}{2}-4x+x-1-1=0$
$\frac{x^3}{2}-3x-2=0.$

2. Clear the fraction to obtain integer coefficients.

Multiply throughout by $2$ :
$x^3-6x-4=0.$

Answer

$x^3-6x-4=0$

AI analysis — may contain mistakes.

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