Package of problems

## Problem

The roots of quadratic polynomials can be nice, integer values. For example $x^2+4x+3$ has $x=-3$ as a root. However, this is not always the case. You will have encountered many quadratic polynomials with roots that are fractions or even irrational numbers.

Convince yourself that $x=\sqrt{2}$ is a root of the quadratic equation $x^2-2=0$ and that $x=\sqrt{3}$ is a root of the quadratic equation $x^2-3=0$.

• Can you find a quadratic polynomial with integer coefficients which has $x=1+\sqrt{2}$ as a root?

What is the other root of this polynomial?

• What if instead $x=1+\sqrt{3}$ is a root? What would the quadratic polynomial be now?

What would the other root be this time?

• Can you generalise your answers to the case where $1+\sqrt{n}$ is a root?

What about the case where $m+\sqrt{n}$ is a root?

If you think you need to revisit methods of solving quadratic equations you may like to look at these resources: Quadratic grids and Factorisable quadratics.