Solve me!
Can you use numerical methods to solve this equation to 1 decimal place?
Age
16 to 18
Challenge level
Problem
Find a solution to this equation to 1 dp.
$$2x^3+34 x^2+567x +8901=0$$
Are there any others?
Did you know ... ?
Numerical solution of equations forms an important part of real-world mathematics and mathematics applied to science, where equations are often too complex to be solved exactly. Mathematicians have developed many advanced techniques for the numerical solution and exploration of equations.
Numerical solution of equations forms an important part of real-world mathematics and mathematics applied to science, where equations are often too complex to be solved exactly. Mathematicians have developed many advanced techniques for the numerical solution and exploration of equations.
Getting Started
This problem draws together ideas of numerical solution of
equations and calculus typically found at the start of a post-16
course.
Student Solutions
Any sensible numerical method will lead to a solution $-16.3(2)$.
In particular, an interval-halving method is efficient and simple to implement. You can make it a little quicker by choosing a sensible starting point: note that any solution would have to be negative, since all of the coefficients are positive; another moment of inspection will also show that the solution must lie between $-10$ and $-100$, giving you a sensible starting point for a computation.
To determine whether there are any other solutions, note that the expression is a cubic and will therefore have either $1$ or $3$ real solutions.
To explore the properties of the cubic $y=2x^3+34x^2+567x+8901$, look at the turning points. Differentiating, we find that
$$
\frac{dy}{dx} = 6x^2+68x+567
$$
The discriminiant of this quadratic is $68^2-4\times 6 \times 567 = -8984$. Since this is negative, there are no turning points and the cubic consequently only has $1$ real solution.