Quadratic Equations Questions - JEE

Let $-\dfrac{\pi}{6} \beta_1$ and $\alpha_2 > \beta_2$, then $\alpha_1 + \beta_2$ equals

If $(1 - p)$ is a root of quadratic equation $x^2 + px + (1 - p) = 0$, then find its roots.

If the quadratic polynomial, $y = (\cot\alpha)x^2 + 2(\sqrt{\sin\alpha})x + \dfrac{1}{2}\tan\alpha$, $\alpha \in [0, 2\pi]$ can take negative values for all $x \in \mathbb{R}$, then the value of $\alpha \in (\pi\lambda, \pi)$, then find the value of $\lambda$.

If $a, b, c, d \in \mathbb{R}$, then the equation $(x^2 + ax - 3b)(x^2 - cx + b)(x^2 - dx + 2b) = 0$ has

Given inequality is $-3 \leq \dfrac{x^2 - \lambda x - 2}{x^2 + x + 1} \leq 2$. Find the number of integral values of $\lambda$.

Given that $a > 0$, $|ax^2 + bx + c| \leq 1$ if $-1 \leq x \leq 1$, $a, b, c \in \mathbb{R}$ and $ax + b$ has its maximum value $2$ when $-1 \leq x \leq 1$. Then $c =$

If the expression $x^2 + 2(a + b + c)x + 3(bc + ca + ab)$ is a perfect square, then

The quadratic polynomial $p(x)$ has the following properties:$p(x)$ can be positive or zero for all real numbers$p(1) = 0$ and $p(2) = 2$Then find the quadratic polynomial.

If $l, m, n$ are the three positive roots of the equation $x^3 - ax^2 + bx - 48 = 0$, then the minimum value of $\dfrac{1}{l} + \dfrac{2}{m} + \dfrac{3}{n}$ equals

If α, β are the nonzero roots of $ax^2 + bx + c = 0$ and $\alpha^2$, $\beta^2$ are the roots of $a^2x^2 + b^2x + c^2 = 0$, then a, b, c are in

Let the equation $ax^2 - bx + c = 0$ has 2 distinct roots in the interval $(0, 1)$ where $a, b, c \in \mathbb{N}$. If $\lambda \leq \log_5(abc)$ for all choices of natural numbers $a, b, c$, then non-negative integral values of $\lambda$ can be:

How many real solutions does the equation $x^7+14x^5+16x^3+30x-560=0$ have?

Find the value of $2 + \dfrac{1}{2 + \dfrac{1}{2 + \dfrac{1}{2+\cdots\infty}}}$.

Let $f(x) = x^2 + bx + c$, where $b, c \in \mathbb{R}$. If $f(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, then the least value of $f(x)$ is

Given that f is a quadratic function such that f(f(1)) = 0 and f(f(2)) = 0. If the quadratic f(x) = x2 + αx + β, find the value of 2|f(0)|.

For the equation $\sqrt{3x^2 + x + 5} = x - 3$, squaring both sides gives $2x^2 + 7x - 4 = 0$, so $x = \frac{1}{2}$ or $x = -4$. Checking both roots in the original equation:

If α, β are the roots of ax2 + c = bx, then the equation (a + cy)2 = b2y in y has the roots

Find the largest natural number $a$ for which the maximum value of $f(x) = a - 1 + 2x - x^2$ is smaller than the minimum value of $g(x) = x^2 - 2ax + 10 - 2a$.

If the equation $x^2 - 3px + 2q = 0$ and $x^2 - 3ax + 2b = 0$ have a common root and the other roots of the second equation is the reciprocal of the other roots of the first, then $(2q - 2b)$ is

Solve $|x^2 + 4x + 3| = x + 1$.

The integral values of $m$ for which the roots of the equation $mx^2 + (2m-1)x + (m-2) = 0$ are rational are given by the expression [where $n$ is integer]

If $(x-2)^6 + (x-4)^6 = 64$, then equation has

If $\alpha$ and $\beta$ are the roots of the equation $x^2 - mx + 2 = 0$ and $\alpha + \dfrac{1}{\beta}$, $\beta + \dfrac{1}{\alpha}$ are the roots of the equation $x^2 - px + q = 0$, then the value of $2q$ equals:

163. If $a, b, c \in \mathbb{R}$ and $a^2 + b^2 + c^2 + 4 = ab + bc + 2c + 2a$, then roots of $ax^2 + bx + c = 0$ are:

If the roots of the equation $bx^2 + cx + a = 0$ be imaginary, then for all real values of $x$, the expression $3b^2x^2 + 6bcx + 2c^2$ is

If $x = 1 + \dfrac{1}{3 + \dfrac{1}{2 + \dfrac{1}{3 + \dfrac{1}{2\ldots\infty}}}}$, then value of $x$ is

The quadratic equation $x^2 + bx + c = 0$ has distinct roots. If 2 is subtract from each root then result are the reciprocal of the original root. The value of $(b^2 + c^2)$ is:

Let $g(x_2) \leq f(x_1)$ for all $x \in [0,1]$, where $f(x)$ is an increasing function on $[0,1]$ with minimum value $f(0) = -1$. Given $h(x) = x^2 - 2ax + 5$, find the minimum value of $a$ such that $h(x) \leq 0$ for at least one $x \in [1, 2]$.

If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 + 8 = 0$, then find the equation whose roots are $\alpha^2$, $\beta^2$, and $\gamma^2$.

The number of solutions to the equation $2\sqrt{1 + \sqrt{1 + (x+1)\sqrt{1 + (x+2)\sqrt{1 + (x+3)(x+5)}}}} = x$ is:

Given that $a > 0$, $|ax^2 + bx + c| \leq 1$ if $-1 \leq x \leq 1$, $a, b, c \in \mathbb{R}$ and $ax + b$ has its maximum value $2$ when $-1 \leq x \leq 1$. Then $b =$

Consider the equation $x^4 - (k-1)x^2 + (2-k) = 0$. The complete set of possible values of real $k$ for which the equation has four distinct real roots is:

The quadratic $x^2 + ax + b + 1 = 0$ has roots which are positive integers, then $(a^2 + b^2)$ can be equal to

Consider the equation $x^4 - (k-1)x^2 + (2-k) = 0$. The complete set of possible values of real $k$ for which the equation has 3 distinct real roots is:

Consider the equation $x^4 - (k-1)x^2 + (2-k) = 0$. The complete set of possible values of real $k$ for which the equation has 2 distinct real roots is:

If $\frac{1}{\sqrt{\alpha}} + \frac{1}{\sqrt{\beta}} = -\frac{b}{a}$ and $\frac{1}{\sqrt{\alpha}} \cdot \frac{1}{\sqrt{\beta}} = \frac{1}{a}$, and $\alpha_1, \beta_1$ are roots of $x^2 + (b^3 - 3ab)x + a^3 = 0$, then the required roots of the new equation are:

If x2 + px + q = 0 and x2 + qx + p = 0, (p ≠ q) have a common root, show that 1 + p + q = 0. Also, show that their other roots are the roots of the equation x2 + x + pq = 0.

Let $\alpha$ and $\beta$ be the real roots of the equation $x^2 - x(k - 2) + \left(k^2 + 3k + 5\right) = 0$. The maximum value of $\alpha^2 + \beta^2$ is:

For the equation $4x^2 - 16x + c = 0$, find the range of $c$ such that both roots are real and lie in the interval $(1, 3)$.

If the roots of the quadratic equation (4p^2 - p - 5)x^2 - (2p - 1)x + 3p = 0 lie on either side of unity, the number of integral values of p is

If x and y are positive integers such that xy + x + y = 71 and x2y + xy2 = 880, then x2 + y2 is equal to

Let $-\frac{\pi}{6} \leq I \leq \frac{\pi}{12}$. Suppose $r_1$ and $s_1$ are the roots of equation $x^2 - 2x\sec I + 1 = 0$ and $r_2$ and $s_2$ are the roots of equation $x^2 + 2x\tan I - 1 = 0$. If $r_1 > s_1$ and $r_2 > s_2$, then $r_1 + s_2$ equals

If α and β (α ≠ β) are the roots of the equation x2 + bx + c = 0, where c ≠ 0 ≠ b, then

The solution of the equation $\frac{8}{[x]} - \frac{9}{x} + \frac{10}{\{x\}} = 2$ is of the form $\frac{k+1}{k}, k \in \mathbb{N}$ then $k = $ _______. ($[x]$ denotes largest integer less than or equal to $x$, and $\{x\}$ denotes fractional part of $x$)

The value of 'a' so that the equation $x^3 - 6x^2 + 11x + a - 6 = 0$ has exactly three integer solutions is _______.

If f(x) = ax2 − bx + c has two distinct roots α and β, and f(0) and f(1) are of the same sign, with the constraint α(1 − α) ≤ 1/4, find the least value of b.

Remainder when $P(x^5)$ is divided by $P(x) = x^4 + x^3 + x^2 + x + 1$ is _______.

If $a,b,c \in \mathbb{R}, a > 10$ and $(x-a)(x-12) + 2 = (x+b)(x+c)$ for all $x \in \mathbb{R}$ then $|b-c| = $ _______.

For real $a,b,c, a+b+c = 2, a^2 + b^2 + c^2 = 6$ and $a^3 + b^3 + c^3 = 8$ then $(1-a)(1-b)(1-c) = $ _______.

Let $f(x) = x^2 + bx + c, b,c \in \mathbb{R}$. If $f(x)$ is a factor of both $x^4 + 6x^2 + 25$ and $3x^4 + 4x^2 + 28x + 5$, then the minimum value of $f(x)$ is _______.

Quadratic Equations Questions (527)