Write \dfrac1\alpha+\dfrac1\beta using sum and product.

\dfrac{\alpha+\beta}{\alpha\beta}.

Identity for \alpha^3+\beta^3?

(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta).

Sum and product of 2\alpha and 2\beta?

Sum 2(\alpha+\beta); product 4\alpha\beta.

Sum and product of \alpha+k and \beta+k?

Sum \alpha+\beta+2k; product \alpha\beta+k(\alpha+\beta)+k^2.

Sum and product of \alpha^3 and \beta^3?

Sum \alpha^3+\beta^3; product (\alpha\beta)^3.

Do you need to solve the equation to find \alpha^2+\beta^2?

No: use (\alpha+\beta)^2-2\alpha\beta.

Sum and product of rootsEdexcel International A Level Further Maths: Flashcards

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Sum of roots of $ax^2+bx+c=0$?

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Sum of roots of $ax^2+bx+c=0$ ?: $\alpha+\beta=-\dfrac ba$ .
Product of roots of $ax^2+bx+c=0$ ?: $\alpha\beta=\dfrac ca$ .
Write $\alpha^2+\beta^2$ using sum and product.: $(\alpha+\beta)^2-2\alpha\beta$ .
Write $\dfrac1\alpha+\dfrac1\beta$ using sum and product.: $\dfrac{\alpha+\beta}{\alpha\beta}$ .
Write $(\alpha-\beta)^2$ using sum and product.: $(\alpha+\beta)^2-4\alpha\beta$ .
Identity for $\alpha^3+\beta^3$ ?: $(\alpha+\beta)^3-3\alpha\beta(\alpha+\beta)$ .
Equation with roots of sum $S$ and product $P$ ?: $x^2-Sx+P=0$ .
Sum and product of $2\alpha$ and $2\beta$ ?: Sum $2(\alpha+\beta)$ ; product $4\alpha\beta$ .
Sum and product of $\alpha+k$ and $\beta+k$ ?: Sum $\alpha+\beta+2k$ ; product $\alpha\beta+k(\alpha+\beta)+k^2$ .
Sum and product of $\alpha^3$ and $\beta^3$ ?: Sum $\alpha^3+\beta^3$ ; product $(\alpha\beta)^3$ .
For $2x^2-7x+4=0$ , find $\alpha+\beta$ and $\alpha\beta$ .: $\frac72$ and $2$ .
Do you need to solve the equation to find $\alpha^2+\beta^2$ ?: No: use $(\alpha+\beta)^2-2\alpha\beta$ .

Exam questions on Sum and product of roots

The roots of the equation $2x^2-7x+4=0$ are $\alpha$ and $\beta$ .
Find the value of $\dfrac1\alpha+\dfrac1\beta$ .2 marks
The roots of the equation $x^2+6x+7=0$ are $\alpha$ and $\beta$ .
Find a quadratic equation, with integer coefficients, whose roots are $2\alpha$ and $2\beta$ .2 marks
The roots of the equation $3x^2-5x+1=0$ are $\alpha$ and $\beta$ .
Show that $\alpha^2+\beta^2=\frac{19}{9}$ .3 marks

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Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).