Quadratic equations practice · Easy

The solution set of the inequation $\left| {\frac{{2x - 1}}{{x - 1}}} \right| > 2,$ is

${\left( {{{\log }_5}x} \right)^2} + \left( {{{\log }_5}x} \right) < 2,$ then $x$ belongs to the interval

If $a,\;b$ and $c$ are in geometric progression and the roots of the equation $a{x^2} + 2bx + c = 0$ are $\alpha$ and $\beta$ and those of $c{x^2} + 2bx + a = 0$ are $\gamma$ and $\delta$, then

The sum of all real roots of the equation $|x-2|^2+|x-2|-2=0$ is

If ${e^{\cos x}} - {e^{ - \cos x}} = 4,$ then the value of $\cos x$ is

If $\alpha, \beta$ are roots of $9 x^2-11 x+1=0$ then value of $\frac{1}{(9 \alpha-11)^2}+\left(\frac{11 \beta-1}{9}\right)$ is-

If $\alpha + \beta = 4$ and ${\alpha ^3} + {\beta ^3} = 44$, then $\alpha ,\;\beta$ are the roots of the equation

If $a,c \ne 0$ and $\alpha ,\beta$ are the roots of the equation $a{x^2} + bx + c = 0$, then the quadratic equation with $1/\alpha$ and $1/\beta$ as its root is

$\frac{x+4}{x-3} < 2$ is satisfied when $x$ satisfies

If one root of the quadratic equation $a{x^2} + bx + c = 0$ is equal to nth power of the other root, then the value of ${\left( {a{c^n}} \right)^{1/\left( {n + 1} \right)}} + {\left( {{a^n}c} \right)^{1/\left( {n + 1} \right)}}$ is equal to

Let $\alpha ,\beta$ be the roots of ${x^2} + bx + 1 = 0.$ Then the equation whose roots are $- \left( {\alpha + \frac{1}{\beta }} \right)$ and $- \left( {\beta + \frac{1}{\alpha }} \right),$ is

Suppose that two persons $A$ and $B$ solve the equation ${x^2} + ax + b = 0$. While solving $A$ commits a mistake in the coefficient of $x$ was taken as 15 in place of $- 9$ and finds the roots as $- 7$ and $- 2$. Then the equation is

If the equations $x^{2}+a x+b=0$ and $x^{2}+a^{\prime} x+b^{\prime}=0$ have a common root, then this common root is equal to

The value of '$k$' for which one of the roots of ${x^2} - x + 3\;k = 0,$ is double of one of the roots of ${x^2} - x + k = 0$ is

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

The value of
$x = 4 + \frac{1}{{5 + \frac{1}{{4 + \frac{1}{{5 + \frac{1}{{4 + ...\infty }}}}}}}}$

If $a < b < c < d$, then the roots of the equation $\left( {x - a} \right)\left( {x - c} \right) + 2\left( {x - b} \right)\left( {x - d} \right) = 0$ are

Two students while solving a quadratic equation in $x$, one copied the constant term incorrectly and got the roots 3 and 2. The other copied the constant term coefficient of ${x^2}$ correctly as $- 6$ and 1 respectively the correct roots are

The sum of all values of $x$, so that $16^{\left(x^2+3 x-1\right)}=8^{\left(x^2+3 x+2\right)}$, is

If $\alpha$ and $\beta$ are the roots of the equation $x^2+p x+q=0$, then the value of $\left(\omega \alpha+\omega^2 \beta\right)\left(\omega^2 \alpha+\omega \beta\right)$, where $\omega$ is an imaginary cube root of unity, is

The value of $x=\sqrt{2+\sqrt{2+\sqrt{2+\ldots \ldots}}}$ is;

If $\alpha$ and $\beta$ are the distinct roots of the equation ${x^2} + {(3)^{1/4}}x + {3^{1/2}} = 0$ then the value of ${\alpha ^{96}}\left( {{\alpha ^{12}} - 1} \right) + {\beta ^{96}}\left( {{\beta ^{12}} - 1} \right)$ is equal to

If the equation $5 x^{2}-10 x+\log _{1 / 5} a=0$ has real roots, then the minimum value of $a$ is

Let $f(x)=\int_1^x \sqrt{\left(2-t^2\right)} d t$, the real roots of the equation $x^2-f^{\prime}(x)=0$ are

If $a, b \in R, a \neq 0$ and the quadratic equation $a x^{2}-b x+1=0$ has imaginary roots, then $a+b+1$ is

The product of the roots of the equation $x^{\log _{10} x}=\frac{10^4}{x^3}$, is

If $\alpha$ and $\beta$ are the roots of the equation $a x^2+b x+c=0$, then roots of

$a x^2-b x(x-1)+c(x-1)^2=0$ (1)

are

A real root of the equation $\log _4\left\{\log _2(\sqrt{x+8}-\sqrt{x})\right\}=0$ is:

The number of integral values of $m$ for which the equation $(1 + {m^2}){x^2} - 2(1 + 3m)x + (1 + 8m) = 0,$ has no real root is

The number of integral solutions of $\frac{{x + 2}}{{{x^2} + 1}} > \frac{1}{2}$ is