Quadratic Equations Questions - JEE

If α, β are the roots of $x^2 - 3x + 1 = 0$, then the equation whose roots are $\frac{1}{\alpha + 2}$, $\frac{1}{\beta + 2}$, is

If the roots of $(a^2 + b^2)x^2 - 2(bc + ad)x + c^2 + d^2 = 0$ are equal, then

Solve the equation $x(x+2)(x^2-1)=-1$.

Number of real values of $\lambda$ such that $(\lambda^2 - 4\lambda + 3)x^2 + (\lambda^2 - 5\lambda + 6)x + (\lambda^2 - 9) = 0$ has more than 2 roots is:

For $x\in\mathbb{R}$, the expression $\dfrac{x^2+2x+c}{x^2+4x+3c}$ can take all real values if $c\in$

If $a(p+q)^2 + 2bpq + c = 0$ and $a(p+r)^2 + 2bpr + c = 0$ $(a \neq 0)$, then

If $\alpha, \beta, \gamma$ are the roots of $x^3 - x^2 - 1 = 0$, then the value of $\dfrac{1+\alpha}{1-\alpha} + \dfrac{1+\beta}{1-\beta} + \dfrac{1+\gamma}{1-\gamma}$ is equal to

The smallest positive integral value of a for which the greater root of the equation $x^2 - (a^2 + a + 1)x + a(a^2 + 1) = 0$ lies between the roots of the equation $x^2 - a^2x - 2(a^2 - 2) = 0$, is less than:

73. Given $x, y \in \mathbb{R}$, $x^2 + y^2 > 0$. Then the range of $\dfrac{x^2 + y^2}{x^2 + xy + 4y^2}$ is

For Problems 13–15Suppose $f(x)$ is a function satisfying the following conditions:(i) $f(0) = 2,\ f(1) = 1$,(ii) $f$ has a minimum value at $x = 5/2$,(iii) For all $x$,\[f'(x) = \begin{vmatrix} 2ax & 2ax-1 & 2ax+b+1 \\ b & b+1 & -1 \\ 2(ax+b) & 2ax+2b+1 & 2ax+b \end{vmatrix}\]$f(x) = 0$ has

If $x^2 - x\sin 2\theta + 2\cos^2\theta = (x-\alpha)(x-\beta)$, then the maximum value of $(2-\alpha)(2-\beta)$ is $5 + \sqrt{a}$. Find $a$.

The equation $(x^2 + x + 1)^2 + 1 = (x^2 + x + 1)(x^2 - x - 5)$ for $x \in (-2, 3)$ will have number of solutions.

If $x^2 + 3x + 1 + \lambda(x+1) > -10$ for all $x \in \mathbb{R}$, and $\lambda$ is an integer, find the sum of all integer values of $\lambda$.

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:

If α, β are the roots of the equation $x^2 - 2x + 3 = 0$. Then the equation whose roots are $P = \alpha^3 - 3\alpha^2 + 5\alpha - 2$ and $Q = \beta^3 - \beta^2 + \beta + 5$ is

Find the number of positive integral values of $k$ for which $kx^2+(k-3)x+1<0$ for at least one positive $x$.

The number of values of k for which $[x^2 - (k-2)x + k^2] \times [x^2 + kx + (2k-1)]$ is a perfect square is

If $b_1 b_2 = 2(c_1 + c_2)$, then at least one of the equations $x^2 + b_1 x + c_1 = 0$ and $x^2 + b_2 x + c_2 = 0$ has

86. If the equation $x^2 + ax + b = 0$ has distinct real roots and $x^2 + a|x| + b = 0$ has only one real root, then which of the following is true?

Find the number of quadratic equations with real roots which remain unchanged even after squaring their roots.

If both roots of the quadratic $ax^2 + bx + c = 0$ lie in $(0, 2)$ then $25ac + 20bc + 16c^2$ is always

What is the minimum height of any point on the curve $y = x^2 - 4x + 6$ above the $x$-axis?

The value of the expression $x^4 - 8x^3 + 18x^2 - 8x + 2$, when $x = 2 + \sqrt{3}$, is

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:

Quadratic Equations Questions (527)