<s>[INST] Find the roots of quadratic polynomial given one root of another quadratic polynomial? if $a,b,c$ are Real numbers and $1$ is a root of equation $ax^2+bx+c=0$ then curve $y = 4ax^2+3bx+2c$ , (a is not zero) intersects $x$ axis at how many points?
I get a relation $a+b+c = 0$ I tried substituting this in given polynomial and tried finding its discriminant but cannot figure out how to find whether it is positive, negative or zero? can someone help?
 [/INST] Since $c=-a-b$, the discriminant $\Delta$ is
$$\begin{align}\Delta&=(3b)^2-4\cdot 4a\cdot 2c\\&=9b^2-32a(-a-b)\\&=32a^2+32ab+9b^2\\&=32a^2+32ab+8b^2+b^2\\&=8(4a^2+4ab+b^2)+b^2\\&=8(2a+b)^2+b^2\\&\gt 0\end{align}$$
because $\Delta=0\iff 2a+b=b=0\iff a=b=0$, which contradicts $a\not=0$.
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