Forming Quadratic Equations when Given Roots Simplified Revision Notes for Leaving Cert Mathematics

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Forming Quadratic Equations when Given Roots

Introduction

Any quadratic can be formed from its roots, the same applies for complex roots.

Conjugate Root Theorem

In the previous observe that the roots we found are conjugates of each other, this applies to all quadratic equations with complex roots, known as the conjugate root theorem.

infoNote

If all the coefficients of a quadratic are real, then the roots are both real or are complex conjugates of each other.

If you are told that $(-3-2i)$ is a root of a quadratic equation, you already know that $(-3+2i)$ is the remaining root.

Sum / Product Property

Any quadratic equation can be formed from its roots by using $z^2-Sz+P=0$ .

$S$ is the sum of the roots.
$P$ is the product of the roots. Example

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$-2+5i$ is a root of a quadratic equation, find the equation.

We know the other root has to be the conjugate :

-2+5i,-2-5i

Derive $S$ and $P$ , the sum and product of the roots respectively.

\begin{align*} S &= (-2+5i)+(-2-5i) \\ &= -4 \\ P &= (-2+5i)\cdot(-2-5i) \\ &= 4+10i-10i-25i^2 \\ &= 4-25(-1) \\ &= 29 \end{align*}

Derive the quadratic equation :

\begin{align*} z^2-Sz+P&=0 \\ z^2 -(-4)z+(29)&=0 \\ z^2+4z+29&=0 \end{align*}

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