the roots of the equation $ax^2 +bx+c=0$ are $\alpha$ and $\beta$ find an equation with roots $\alpha + \beta$ and $\alpha \beta$.

Question

The roots of the equation $ax^2 +bx+c=0$ where $a,b,c \in\Bbb Z^+$ are $\alpha$ and $\beta$. Find a quadratic with integer coefficients whose roots are $\alpha + \beta$ and $\alpha \beta$.

So my workings are below but from graphing a few curves with my equation, it hasn't seemed to work. Any help would be great.

$$\alpha + \beta = \frac {-b} {a} $$ $$\alpha \beta = \frac c a$$

So the new equation will need to have roots $\frac {-b} {a} $ and $\frac c a$. so the equation can be written in the form $(x-p)(x-q)=0$ meaning the new equation is $$(x+\frac {b} {a})(x-\frac c a) = x^2 +(\frac b a - \frac c a )x - \frac {bc}{a^2}=0$$ $$a^2x+a(b-c)x-bc=0$$ This final answer doesn't seem to yield me the correct result when graphing, any help would be great.

When graphing it, it didn't seem to work. Is my logic correct? — H.Linkhorn, Feb 22 '19 at 19:13
Of course I may be overlooking something, but I cannot see an obvious error. – Can you provide a concrete example where your result is wrong? — Martin R, Feb 22 '19 at 19:14
so if you take $x^2 +3x-10$ which has roots -5 and 2. the equation I've got gives $x^2+13x-30$ which isn't right as it has roots 2 and -15. neither of which are result of summation or multiplication. — H.Linkhorn, Feb 22 '19 at 19:17
Your method gives $x^2+13x+30$, having the roots $-3$ and $-10$. — Martin R, Feb 22 '19 at 19:20
Ok, maybe I've just made a sign error in my workings somewhere then — H.Linkhorn, Feb 22 '19 at 19:21
@H.Linkhorn Yeah, it's $-bc/a^2$, and since $c$ is negative, it's $+30$. — Infiaria, Feb 22 '19 at 19:22

Bernard · Answer 1 · 2019-02-22T19:22:00.450

1

It is a high-school theorem that, $s$ and $p$ being given, if the sum of two numbers (not necessarily distinct) is $s$ and their product is $p$, these numbers are the roots of the quadratic equation: $$x^2-sx+p=0.$$ So your answer is perfectly correct.

edited Feb 22 '19 at 19:22

answered Feb 22 '19 at 19:18

Bernard

175,478

I know this, and have used it within my workings but I want an equation which has roots $\alpha + \beta$ and $\alpha \beta$ given the original has roots $\alpha$ and $\beta$ – H.Linkhorn Feb 22 '19 at 19:20
You use this theorem, after you've calculated the sum and the product with Vieta's relations. – Bernard Feb 22 '19 at 19:23

the roots of the equation $ax^2 +bx+c=0$ are $\alpha$ and $\beta$ find an equation with roots $\alpha + \beta$ and $\alpha \beta$.

1 Answers1