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Shades of Fermat's Last Theorem

The familiar Pythagorean 3-4-5 triple gives one solution to (x-1)^n + x^n = (x+1)^n so what about other solutions for x an integer and n= 2, 3, 4 or 5?

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Exhaustion

Find the positive integer solutions of the equation (1+1/a)(1+1/b)(1+1/c) = 2

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Code to Zero

Find all 3 digit numbers such that by adding the first digit, the square of the second and the cube of the third you get the original number, for example 1 + 3^2 + 5^3 = 135.

In Between

Stage: 5 Challenge Level: Challenge Level:2 Challenge Level:2

Why do this problem?

This is a non standard example of a quadratic inequality, the solution of which will involve algebraic manipulation. It can be used to help learners to practise the skills of estimation and approximation prior to engaging in an algebraic solution.

Possible approach

There are various ways in which this problem can be tackled. It can obviously be set 'straight' to learners, or used in a slightly wider context, as described here.

To begin with, can learners assess a rough range of values that is likely to emerge upon solving the inequality? For example, would very large or very small values of $x$ satisfy the inequality? As a class, who can spot the largest or smallest values which will satisfy the inequality. This might naturally lead to a numerical investigation, although in this case it turns out that there is an exact algebraic solution to the problem.

Once the problem has some sense of a numerical estimation, encourage the class to move on to an algebraic solution. Learners will soon discover that they need to deal with the awkward $\sqrt{x}$ terms. One way is to square both sides of the inequality; another is to make the substitution $\sqrt{x} = p$. The inequality can then be rearranged into a quadratic inequality.

It is important for learners not just to blindly apply rules of algebraic manipulation to the resulting inequalities. For each manipulation (squaring, rearranging etc.) learners should explain clearly why the operation preserves the inequality sign.

The next step will be to try to solve the quadratic inequality. Learners might use an algebraic approach or a graphical approach, but in each case will be required to use the quadratic equation formula to find the solutions. It may be helpful for the class to work together to solve the inequality. A fruitful class discussion, where the learners are able to volunteer suggestions for the steps in the working, is more likely to ensue if they have at least started solving the problem for themselves and have worked through the earlier steps.
Finally, once a solution is found it is good practice to check that the boundaries of the inequality work and also to compare these to the original estimates.

Key questions

Can you estimate an approximate range of values for which the inequality is satisfied?

What things might we try to remove the awkward $\sqrt{x}$ term?

How do we find the factors of a quadratic expression?

If you know that the product of two factors is negative what can you say about the factors?

Possible extension

Can you make up a similar inequality which has solutions $3< x < 5$?. How about $a < x < b$?

Possible support

Explicity suggest that learners substitute $\sqrt{x}= p$.