Edexcel A-Level Mathematics AS Fp1 2 Roots of Quadratic Equations Questions

Practise using sums and products of quadratic roots to transform expressions and form new equations without solving for the roots.

Exam points

use α+β and αβ from a quadratic to calculate transformed root expressions
apply identities for α²+β² or α³+β³ before substituting known root sums and products
form a new quadratic by finding the sum and product of its transformed roots

Covered syllabus points

FP1.2.1 - Sum of roots and product of roots FP1.2.2 - Manipulation of expressions FP1.2.3 - Forming quadratic equations with 1 1 1 1 2

[Maximum number: 9]

The quadratic equation

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

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

(a)

write down the value of $(\alpha+\beta)$ and the value of $\alpha \beta$

[ 1 ]

$2 x^{2}-3 x+7=0$ $\alpha+\beta=\frac{3}{2}, \quad \alpha \beta=\frac{7}{2}$
(1)

(b)

determine the value of $\alpha^{2}+\beta^{2}$

[ 2 ]

$\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2 \alpha \beta$ $=\left(\frac{3}{2}\right)^{2}-2\left(\frac{7}{2}\right)=-\frac{19}{4}$
(2)

(c)

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.

[ 6 ]

For roots $\alpha-\frac1{\beta^2}$ and $\beta-\frac1{\alpha^2}$ :
$\text{Sum}=\alpha+\beta-\frac{\alpha^2+\beta^2}{\alpha^2\beta^2} =\frac32-\frac{-19/4}{(7/2)^2}=\frac{185}{98}.$ $\text{Product}=\alpha\beta-\frac{\alpha+\beta}{\alpha\beta}+\frac1{\alpha^2\beta^2} =\frac72-\frac37+\frac4{49}=\frac{309}{98}.$ $x^2-\frac{185}{98}x+\frac{309}{98}=0.$ $98x^2-185x+309=0.$

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Covered syllabus points

Question 1

Question (a)

Question (b)

Question (c)