Question

If all values of $a$ for which the equation ${4x}^2-2x+a=0$ has two roots lie in the interval $(-1,1)$ is the set $(c,b)$, then $-\frac{c}{b}$ is equal to

Answer 1

Let $f\left(x\right)={4x}^2-2x+a$ as both roots of the equation $f\left(x\right)=0$ are lie between $(-1,1)$ we can take D $\ge$0, $af(-1)>0$ and $-1<\frac{1}{4}<1$.
I. consider D $\ge$ 0: ${(-2)}^2-4.4a\ge 0\Rightarrow a\le \frac{1}{4}$ .............(1)
II.Consider $af(-1)>0$
$4(4+2+a)>0$
$a>-6$ $\Rightarrow aϵ(-6,\infty )$ ..........(2)
III consider $af\left(1\right)>0$
$4(4-2+a)>0$
$a>-2$ $\Rightarrow aϵ(-2,\infty )$......... (3)
Hence, the values of satisfying (1), (2) and (3) at the same time are $aϵ(-2,\frac{1}{4})$