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

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.

• Why do you think that the result is wrong? – Martin R Feb 22 at 19:13
• When graphing it, it didn't seem to work. Is my logic correct? – H.Linkhorn Feb 22 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 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 at 19:17
• Ok, maybe I've just made a sign error in my workings somewhere then – H.Linkhorn Feb 22 at 19:21

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.
• 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 at 19:20