Quadratic Equations

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $(\alpha+\beta)(\beta+\gamma)(\gamma+\alpha)=$

If $\alpha, \beta$ are the roots of $x^2-5 \gamma x-6 \delta=0$ and $\gamma, \delta$ are the roots of $x^2-5 \alpha x-6 \beta=0$, then $\alpha+\beta+\gamma+\delta=$

If $\alpha$ is the common root of the quadratic equations $x^2-5 x+4 a=0, x^2-2 a x-8=0$, where $a \in R$, then the value $\alpha^4-\alpha^3+68$ is

If $x^2-4 a x+5+a>0$ for all $x \in R$ whenever $a \in(\alpha, \beta)$, then $4 \beta+\alpha=$

If the area of a square is 575 square units, then the approximate value of its side is

The polynomial equation of degree 5 whose roots are the roots of the equation $x^5-3 x^4-x^3+11 x^2-12 x+4=0$ each increased by 2 , is

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3-12 x^2+k x-18=0$ and one of them is thrice the sum of the other two roots, then $\alpha^2+\beta^2+\gamma^2-k=$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $5 x^3-4 x^2+3 x-2=0$, then $\alpha^3+\beta^3+\gamma^3=$

After the roots of the equation $6 x^3+7 x^2-4 x-2=0$ are diminished by $h$, if the transformed equation does not contain $x$ term, then the product of all the possible value of $h$ is

The product of all the real roots of the equation $|x|^2-5|x|+6=0$

If the difference of the roots of the equation $x^2-7 x+10=0$ is same as the difference of the roots of the equation $x^2-17 x+k=0$, then a divisor of $k$ is $x^2-7 x+10=0$

The number of distinct quadratic equations $a x^2+b x+c=0$ with unequal real roots that can be formed by choosing the coefficients $a, b, c(a \neq b \neq c)$ from the set $\{0,1,2,4\}$ is

The set of all real values of $x$ for which $\frac{x^2-1}{(x-4)(x-3)} \geq 1$ is

The number of solutions of the equation $\sqrt{3 x^2+x+5}=x-3$ is

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $2 x^3+3 x^2-5 x-7=0$, then $\frac{1}{\alpha^2}+\frac{1}{\beta^2}+\frac{1}{\gamma^2}=$

Two roots of the equation, $a x^4+b x^3+c x^2+d x+e=0$ are positive and equal. If the product of the other two real roots is 1 , then

If the sum of two roots of the equation $x^4+2 x^3-7 x^2-8 x+12=0$ is zero, then the sum of the squares of the other two roots is

If $\alpha, \beta, \gamma$ are the roots of the equation,
$$ \begin{aligned} & x^3+a x^2+b x+c=0, \text { then }(\alpha+\beta-2 \gamma) \\ & (\beta+\gamma-2 \alpha)(\gamma+\alpha-2 \beta)= \end{aligned} $$

If $\alpha, \beta$ are the roots of the equation $x^2+b x+c=0$ satisfying the conditions $\alpha+\beta=5$ and $\alpha^3+\beta^3=60$, then $3 c+2=$

Let $(a-3) x^2+12 x+(a+6)>0, \forall x \in R$ and $a \in(\ell, \infty)$. If $a$ is the least positive integral value of $a$, then the roots of $(\alpha-3) x^2+12 x+(\ell+2)=0$ are

The cubic equation whose roots are the squares of the roots of the equation $x^3-2 x^2+3 x-4=0$ is

If the roots of the equation $x^2+2 a x+b=0$ are real, distinct and differ atmost by 2 m , then $b$ lies in the interval

If $\alpha, \beta, \gamma$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $\alpha^3+\beta^3+\gamma^3=$

If $(2 k-1) x^2-2(3 k-2) x+4 k>0$ for every $x \in R$, then the sum of all possible integral values of $k$ is

If $\alpha$ is a repeated root of multiplicity 2 of the equation $18 x^3-33 x^2+20 x-4=0$, then

The equation $6 x^4-5 x^3+13 x^2-5 x+6=0$ will have

The set of all real values of $x$ satisfying the inequation $\frac{8 x^2-14 x-9}{3 x^2-7 x-6}>2$ is

Let ' $a$ ' be a non-zero real number. If the equation whose roots are the squares of the roots of the cubic equation $x^3-a x^2+a x-1=0$ is identical with this cubic equation, then ' $a$ ' =

When the roots of $x^3+\alpha x^2+\beta x+6=0$ are increased by 1 , if one of the resultant values is the least root of $x^4-6 x^3+11 x^2-6 x=0$, then

If $\alpha \neq 0$ and zero are the roots of the equation $x^2-5 k x+\left(6 k^2-2 k\right)=0$, then $\alpha=$

$f(x)$ is a quadratic polynomial satisfying the condition $f(x)+f\left(\frac{1}{x}\right)=f(x) f\left(\frac{1}{x}\right)$. If $f(-1)=0$, then the range of $f$ is

If $x^2-5 x+6$ is a factor of $f(x)=x^4-17 x^3+k x^2-247 x+210$, then the other quadratic factor of $f(x)$ is

If $a x^2+b x+c<0 \forall x \in R$ and the expressions $c x^2+a x+b$ and $a x^2+b x+c$ have their extreme values at the same point $x$, then for the expression $c x^2+a x+b$

Given $f(x)=x^2-5 x+4$. Out of first 20 natural numbers, if a number $x$ is chosen at random, then the probability that the chosen $x$ satisfies the inequality $f(x)>10$ is

The roots $\alpha, \beta$ of the equation $x^2-6(k-1) x+4(k-2)=0$ are equal in magnitude but opposite in sign, if $\alpha>\beta$, then the product of the roots of the equation $2 x^2-\alpha x+6 \beta(\alpha+1)=0$

If $\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3-13 x^2+k x+189=0$ such that $\beta-\gamma=2$, then $\beta+\gamma: k+\alpha=$

All the values of $k$ such that the quadratic expression $2 k x^2-(4 k+1) x+2$ is negative for exactly three integrals values of $x$, lie in the interval

If the harmonic mean of the roots of the equation $\sqrt{2} x^2-b x+(8-2 \sqrt{5})=0$ is

The cubic equation whose roots are the square of the roots of the equation is
\(12 x^3-20 x^2+x+3=0\)

The set of all values of $k$ for which the inequality $x^2-(3 k+1) x+4 k^2+3 k-3>0$ is true for all real values of $x$, is

$\alpha, \beta$ and $\gamma$ are the roots of the equation $x^3+3 x^2-10 x-24=0$ If $\alpha(\beta+\gamma), \beta(\gamma+\alpha)$ and $\gamma(\alpha+\beta)$ are the roots of the equation $x^3+p x^2+q x+r=0$, then $q$ is equal to

The set of all real values of $x$ satisfying the inequality $\frac{7 x^2-5 x-18}{2 x^2+x-6}<2$ is

If $\alpha$ and $\beta$ are two distinct negative roots of $x^5-5 x^3+5 x^2-1=0$, then the equation of least degree with integer coefficients having $\sqrt{-\alpha}$ and $\sqrt{-\beta}$ as its roots, is

If both the roots of the equation $x^2-6 a x+2-2 a+9 a^2=0$ exceed 3 , then

The quotient, when $3 x^5-4 x^4+5 x^3-3 x^2+6 x-8$ is divided by $x^2+x-3$ is

The set of all real values ' $a$ ' for which $-1<\frac{2 x^2+a x+2}{x^2+x+1}<3$ holds for all real values of $x$ is

If ' $a$ ' is a rational number, then the roots of the equation $x^2-3 a x+a^2-2 a-4=0$ are

Roots of the equation $a(b-c) x^2+b(c-a) x+c(a-b)=0$ are

The algebraic equation of degree 4 whose roots are translate of the roots of the equation. $x^4+5 x^3+6 x^2+7 x+9=0$ by -1 is

The equation $x^4-x^3-6 x^2+4 x+8=0$ has two equal roots. If $\alpha, \beta$ are the other two roots of this equation, then $\alpha^2+\beta^2=$