Embibe Experts Solutions for Chapter: Quadratic Equations, Exercise 4: Exercise-4

The range of values for which the quadratic expression $a{x}^2+(a-2)x-2$ is negative for exactly two integral values of $x$ is $[p, q)$. Find $p+q$.

Find the number of real roots of ${\left(x+\frac{1}{x}\right)}^3+\left(x+\frac{1}{x}\right)=0$.

If $\alpha ,\beta$ are roots of the equation $x^2-34x+1=0,$ evaluate $\left|\sqrt[4]{\alpha }-\sqrt[4]{\beta }\right|,$ where $\sqrt[4]{·}$ denotes the principal value.

The range of values of $a$ for which the equation ${\left(x^2+x+2\right)}^2-(a-3)\left(x^2+x+2\right)\left(x^2+x+1\right)+(a-4){\left(x^2+x+1\right)}^2=0$ has at least one real root is $\left[p, q\right]$. Find $2p+3q$.

Find the sum of the integral values of $a$ for which the equation $x^4-\left(a^2-5a+6\right)x^2-\left(a^2-3a+2\right)=0$ has only real roots.

If $\mathrm{\alpha }, {\mathrm{\beta }}^2$ are integers, ${\mathrm{\beta }}^2$ is non-zero multiple of $3$ and $\mathrm{\alpha }+i\beta , -2\mathrm{\alpha }$ are roots of $x^3+a{x}^2+bx-316=0$, $a, b, \beta \in R$ then find $a, b$.

Let polynomial $f\left(x\right)=a{x}^4+b{x}^3+c{x}^2+dx+e$ have integral coefficient (where $a>0$). If there exist four distinct integers ${\alpha }_1, {\alpha }_2, {\alpha }_3, {\alpha }_4 \left({\alpha }_1<{\alpha }_2<{\alpha }_3<{\alpha }_4\right)$ such that $f\left({\alpha }_1\right)=f\left({\alpha }_2\right)=f\left({\alpha }_3\right)=f\left({\alpha }_4\right)=5$ and equation $f\left(x\right)=9$ has integral roots, then find

$\left(\mathrm{i}\right)$ $f\left(\frac{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4}{4}\right)$

$\left(\mathrm{ii}\right)$ $f^{\text{'}}\left(\frac{{\alpha }_1+{\alpha }_2+{\alpha }_3+{\alpha }_4}{4}\right)$

$\left(\mathrm{iii}\right)$ Range of $f\left(x\right)$ in $\left[{\alpha }_2, {\alpha }_3\right]$

$\left(\mathrm{iv}\right)$ Difference of largest and smallest root of equation $f\left(x\right)=9$

If $x$ and $y$ both are non-negative integral values for which ${\left(xy-7\right)}^2=x^2+y^2$, then find the sum of all possible values of $x$.