Solve me!
Can you use numerical methods to solve this equation to 1 decimal place?
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.
Getting Started
This problem is trivial if you use an equation solver on your calculator to solve it or zoom in using graph-drawing software to get as good an approximation as you want. Instead, try to use some reasoning and only the calculating power of a basic scientific calculator to get an approximate solution as efficiently as possible.
Where could you start? What happens when $x$ is positive, $0$ or negative? What do you know about cubic graphs? Would finding any turning points help?
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.
Teachers' Resources
Why do this problem?
This problem draws together ideas of numerical solution of equations and calculus typically found within a post-16 course. As such, it could be used as a consolidation exercise in which students select an appropriate strategy from the techniques they have learned.
Alternatively, it could be used as a motivating problem at the start of a topic on numerical solution of equations.
Possible approach
Students could be challenged to find their own way in to the problem. They should be encouraged to think first rather than diving straight into calculations and will need to be warned off simply using an equation-solving function or graph-drawing software. Perhaps set up a scenario where students use as few calculations as possible, but where calculations are saved by reasoning rather than luck.
Once students have deduced as much as possible without calculators, they could be allowed just one initial value of $x$ substituted into the left-hand side of the equation. Some or all of the results could be shared as a class, ideally with one or more negative results, to allow for discussion of what further deductions can be made. Students could then be asked to devise a strategy before implementing it, keeping track of the number of calculations needed to get to a 1 d.p. level of accuracy. When students believe they have reached that level of accuracy, they should be prompted to show how they can justify their accuracy. Finally various methods can be compared in terms of efficiency and ease of implementation.
Whether students are using this problem for consolidation or to introduce the topic, working in pairs will be useful to encourage discussion. Ideally time should be left to draw together the strands of discussion, emphasising key points about changes in sign, continuity of functions and the justification required for a given level of accuracy.
Key Questions
Where could you start? What happens when $x$ is positive, $0$ or negative?
What do you know about cubic graphs? Would finding any turning points help?
What do we know about the values of a function either side of a solution?
Can you think of functions for which this sort of approach might not work?
Does this method tell us anything about the number of roots of an equation?
Possible extension
Possible support
Suggest that students try specific values such as $-10$ and $-100$. A spreadsheet could be used to speed up calculation while keeping a record of approximations and results.