Mathematics 9709 · AS & A Level · Numerical solution of equations

Numerical solution of equations — practice question

The equation $x^5 - 3x^3 + x^2 - 4 = 0$ has exactly one positive root.
(i)[2]

Confirm by calculation that this root is located between $1$ and $2$.

(ii)[1]

Demonstrate that the equation may be rewritten in the form $x = \sqrt[3]{\left(3x + \frac{4}{x^2} - 1\right)}$.

(iii)[3]

Apply an iterative formula derived from this rearrangement to find the positive root correct to $2$ decimal places. Record each iterate to $4$ d.p.

Worked solution & mark scheme

This 6-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Check the sign of $x^5 - 3x^3 + x^2 - 4$ at $x=1$ and $x=2$, or any equivalent approach” …

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