Irrational Roots
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?
Underground Mathematics is hosted by Cambridge Mathematics. The project was originally funded by a grant from the UK Department for Education to provide free web-based resources that support the teaching and learning of post-16 mathematics.
Visit the site at undergroundmathematics.org to find more resources, which also offer suggestions, solutions and teacher notes to help with their use in the classroom.
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