Pearson Edexcel IAL Mathematics FP1.2 Roots of quadratic equations Question Bank

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 ]

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

B1

(1)

(b)

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

[ 2 ]

$\alpha^{2}+\beta^{2}=(\alpha+\beta)^{2}-2 \alpha \beta$

M1

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

A1

(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 $\beta-\frac1{\alpha^2}$ :
$\text{Sum}=\alpha+\beta-\frac{\alpha^2+\beta^2}{\alpha^2\beta^2} =\frac32-\frac{-19/4}{(7/2)^2}=\frac{185}{98}.$
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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Covered syllabus points

FP1.2 - Roots of quadratic equations question 1

Question (a)

Question (b)

Question (c)