Roots of Quadratic

Algebra Level 3

Consider real numbers $m$ and $n$ such that $-1+2i$ is a root of the quadratic equation $x^2+mx+n=0$ . If $m$ and $n$ are the two roots of the quadratic equation $x^2+Ax+B=0$ , what is the value of $B-A$ ?

11

17

13

15

1 solution

Tom Engelsman
Nov 9, 2020

The two roots in question are complex conjugate pairs $-1 \pm 2i$ , which after applying Vieta's Formulae gives:

$m=-[(-1+2i)+(-1-2i)] = 2$ ;

$n = (-1+2i)(-1-2i) = 5$ .

Now if $m$ and $n$ are the roots to $x^2 + Ax +B = (x-m)(x-n) = (x-2)(x-5) = 0$ , then $A = -7, B = 10 \Rightarrow B-A = \boxed{17}.$

Roots of Quadratic

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