Quadratic Equations

If for some $p,q,r\in\mathbf R$, not all have same sign, one of the roots of the equation $\left(p^2+q^2\right)x^2-2q(p+r)x+q^2+r^2=0$ is also a root of the equation $x^2+2x-8=0$, then $\dfrac{q^2+r^2}{p^2}$ is equal to ______.

The number of distinct real roots of the equation $x^5\left(x^3-x^2-x+1\right)+x\left(3x^3-4x^2-2x+4\right)-1=0$ is ______.

The minimum value of the sum of the squares of the roots of $x^2+(3-a)x+1=2a$ is:

Let the abscissae of the two points P and Q on a circle be the roots of $x^2-4x-6=0$ and the ordinates of P and Q be the roots of $y^2+2y-7=0$. If PQ is a diameter of the circle $x^2+y^2+2ax+2by+c=0$, then the value of $(a+b-c)$ is ______.

For $k\in\mathbf R$, let the solutions of the equation $\cos\left(\sin^{-1}\left(x\cot\left(\tan^{-1}\left(\cos\left(\sin^{-1}x\right)\right)\right)\right)\right)=k$, $0<|x|<\dfrac1{\sqrt2}$ be $\alpha$ and $\beta$, where the inverse trigonometric functions take only principal values. If the solutions of the equation $x^2-bx-5=0$ are $\dfrac1{\alpha^2}+\dfrac1{\beta^2}$ and $\dfrac\alpha\beta$, then $\dfrac b{k^2}$ is equal to ______.

If $\alpha$, $\beta$ are the roots of the equation

$$x^2-\left(5+3^{\sqrt{\log_35}}-5^{\sqrt{\log_53}}\right)x+3\left(3^{(\log_35)^{\frac13}}-5^{(\log_53)^{\frac23}}-1\right)=0,$$

then the equation, whose roots are $\alpha+\dfrac1\beta$ and $\beta+\dfrac1\alpha$, is :

For $p,q\in\mathbf R$, consider the real valued function $f(x)=(x-p)^2-q$, $x\in\mathbf R$ and $q>0$. Let $a_1$, $a_2$, $a_3$ and $a_4$ be in an arithmetic progression with mean p and positive common difference. If $|f(a_i)|=500$ for all $i=1,2,3,4$, then the absolute difference between the roots of $f(x)=0$ is ______.

The sum of all real values of $x$ for which $\dfrac{3x^2-9x+17}{x^2+3x+10}=\dfrac{5x^2-7x+19}{3x^2+5x+12}$ is equal to ______.

Let $\alpha$, $\beta$ be the roots of the equation $x^2-\sqrt2x+\sqrt6=0$ and $\dfrac1{\alpha^2}+1$, $\dfrac1{\beta^2}+1$ be the roots of the equation $x^2+ax+b=0$. Then the roots of the equation $x^2-(a+b-2)x+(a+b+2)=0$ are :

Let $f(x)=ax^2+bx+c$ be such that $f(1)=3$, $f(-2)=\lambda$ and $f(3)=4$. If $f(0)+f(1)+f(-2)+f(3)=14$, then $\lambda$ is equal to :

If $\dfrac1{(20-a)(40-a)}+\dfrac1{(40-a)(60-a)}+\dots+\dfrac1{(180-a)(200-a)}=\dfrac1{256}$, then the maximum value of a is :

Let S = {θ ∈ (0, 2π) : $7\cos^2\theta-3\sin^2\theta-2\cos^22\theta=2$}. Then, the sum of roots of all the equations $x^2-2(\tan^2\theta+\cot^2\theta)x+6\sin^2\theta=0$, θ ∈ S, is _______.

Let α, β (α > β) be the roots of the quadratic equation $x^2-x-4=0$. If $P_n=\alpha^n-\beta^n$, n ∈ ℕ, then $\dfrac{P_{15}P_{16}-P_{14}P_{16}-P_{15}^2+P_{14}P_{15}}{P_{13}P_{14}}$ is equal to _______.

Let A be a matrix and $\begin{bmatrix}1&2\\0&1\end{bmatrix}A\begin{bmatrix}2&0\\-1&1\end{bmatrix}=\begin{bmatrix}1&0\\0&1\end{bmatrix}$. If |A| and |A$^2$| are the roots of the quadratic equation $ax^2+bx+3=0$, then a + b − ab is equal to

The number of solutions of the equation $|x^2+3x+2|+|x+5|-2=0$, x ∈ ℝ, is _______.

The number of integral values of b for which $2x^3-3x^2-12x+b=0$ has three distinct real roots is _______.

For $x \in \mathbb R$, the number of real roots of the equation
$$3x^2 - 4|x^2 - 1| + x - 1 = 0$$
is ____.

Let p and q be two positive numbers such that p + q = 2 and $p^4+q^4=272$. Then p and q are roots of the equation :

Suppose $a, b$ denote the distinct real roots of the quadratic polynomial $x^2 + 20x - 2020$ and suppose $c, d$ denote the distinct complex roots of the quadratic polynomial $x^2 - 20x + 2020$. Then the value of

$$ac(a-c) + ad(a-d) + bc(b-c) + bd(b-d)$$

is

Let $C_1$ and $C_2$ be two biased coins such that the probabilities of getting head in a single toss are $\dfrac23$ and $\dfrac13$, respectively. Suppose $\alpha$ is the number of heads that appear when $C_1$ is tossed twice, independently, and suppose $\beta$ is the number of heads that appear when $C_2$ is tossed twice, independently. Then the probability that the roots of the quadratic polynomial $x^2 - \alpha x + \beta$ are real and equal, is