Edexcel IAL Mathematics FP1.2.3 forming new quadratic equations

Practise forming quadratic equations whose roots are algebraic transformations of α and β, using sums and products only.

Exam points

clear fractions to give the new quadratic in integer-coefficient form
use earlier α²+β² or α³+β³ results when forming the new equation

[Maximum number: 6]

The quadratic equation

2 x^{2}-3 x+7=0

has roots $\alpha$ and $\beta$
Without solving the equation,

find a quadratic equation which has roots

\left(\alpha-\frac{1}{\beta^{2}}\right) \text { and }\left(\beta-\frac{1}{\alpha^{2}}\right)

giving your answer in the form $p x^{2}+q x+r=0$ where p, q and r are integers to be determined.

For roots $\alpha-\frac1{\beta^2}$ and $\beta-\frac1{\alpha^2}$ :
$\text{Sum}=\alpha+\beta-\frac{\alpha^2+\beta^2}{\alpha^2\beta^2} =\frac32-\frac{-19/4}{(7/2)^2}=\frac{185}{98}.$
M1 A1

$\text{Product}=\alpha\beta-\frac{\alpha+\beta}{\alpha\beta}+\frac1{\alpha^2\beta^2} =\frac72-\frac37+\frac4{49}=\frac{309}{98}.$
M1 A1

$x^2-\frac{185}{98}x+\frac{309}{98}=0.$
M1

$98x^2-185x+309=0.$
A1

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FP1.2.3 - Forming quadratic equations with 1 1 1 1 2 question 1