Coefficient of a cubic expression $x^3-2x^2+ax+10=0$ such that sum of two roots is zero

Question

So the given cubic is $x^3-2x^2+ax+10=0$. The condition is one of the root is additive inverse of another. I need to find the coefficient $a$. I did some algebraic calculations and cancelling of the cubic terms and square terms and got answer as $-25$. Just want to know whether its right if not then please explain me how to approach such sums.

Answer 1

Hint: The coefficient of $x^2$ is minus the sum of the three roots and you are given that two of the roots add up to zero ...

Answer 2

Hint

So, you face the situation $$x^3-2x^2+ax+10=(x-p)(x+p)(x+q)=0$$ Put everything on the same side, expand and simplify; this should give $$(q+2) x^2- \left(a+p^2\right)x -(p^2q+10)=0$$ Now, set each coefficient equal to zero.

I am sure that you can take it from here.

Answer 3

As per given condition,let $\alpha, -\alpha$ & $\beta$ be the roots of the given cubic equation: $x^3-2x^2+ax+10=0$ then the sum of roots $$\alpha+(-\alpha)+\beta=-\frac{\text{coefficient of} \ x^2}{\text{coefficient of} \ x^3}$$$$\beta=-\frac{(-2)}{1}=2$$ now, substituting $\beta=2$ in the cubic equation, one should get $$(2)^3-2(2)^2+a(2)+10=0$$ $$2a+10=0$$$$\implies a=\color{red}{-5}$$