<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] If $1$ is a root of $ax^2 + bx + c = 0$ Then 
\begin{align}
    ax^2 + bx + c
        &= a(x-1)(x - \lambda) \\
        &= ax^2 - a(1 + \lambda)x + a\lambda \\
\end{align}
So $b = -a(1+\lambda)$ and $c = a\lambda$.
The descriminant of $4ax^2 + 3bx + 2c$ is
\begin{align}
    9b^2 - 32ac
        &= 9a^2(1+\lambda)^2 -32a^2\lambda \\
        &= a^2(9\lambda^2-14\lambda+9) \\
        &= 9a^2\left[\left(x- \dfrac 79 \right)^2 + \dfrac{32}{9}\right]
\end{align}
Which is positive.
Hence the polynomial has two real roots.
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