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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