Quadratic equations practice · Easy

The roots of the equation $\left( {q - r} \right){x^2} + \left( {r - p} \right)x + \left( {p - q} \right) = 0$ are

If $\alpha$ and $\beta$ are the roots of the equation ${x^2} - ax + b = 0$ and ${A_n} = {\alpha ^n} + {\beta ^n},$ then which one of the following is true?

Let $\alpha$ and $\beta$ be the roots of ${x^2} - 6x - 2 = 0$. If ${a_n} = {\alpha ^n} - {\beta ^n}$ for $n \ge 1$, then the value of $\frac{{{a_{10}} - 2{a_8}}}{{3{a_9}}}$ is

Let $\left( {\sin a} \right){x^2} + \left( {\sin a} \right)x + \left( {1 - \cos a = 0} \right)$. The value of $a$ for which roots of this equation are real and distinct, is

$\sin A,\sin B,\cos A$ are in GP. Roots of ${x^2} + 2x\cot B + 1 = 0\;$ are always

The value of $\lambda$ for which the quadratic equation $3 x^2+2\left(\lambda^2+1\right) x+\left(\lambda^2-3 \lambda+2\right)=0$ has roots of opposite signs, lies in the interval.

The solution set of inequation ${\log _{1/3}}\left( {{2^{x + 2}} - {4^x}} \right) \ge - 2,$ is

The roots of $a{x^2} + 2\;bx + c = 0$ and $b{x^2} - 2\sqrt {ac} \;x + b = 0$ are simultaneously real, then

The value of $p$ for which the quadratic equation $x^2-p x+p+3=0$ has reciprocal roots is

In writing an equation of the form $a{x^2} + bx + c = 0;$ the coefficient of $x$ is written incorrectly and roots are found to be equal. Again in writing the same equation the constant term is written incorrectly and it is found that one root is equal to those of the previous wrong equation while the other is double of it. If $\alpha$ and $\beta$ be the roots of correct equation, then ${\left( {\alpha - \beta } \right)^2}$ is equal to

If ${x_1},{x_2},{x_3}$ are distinct roots of the equation $a{x^2} + bx + c = 0$, then

The numbers of pairs $(a,\,\,b)$ of real numbers, such that whenever $\alpha$ is a root of the equation ${x^2} + ax + b = 0,\,\,\,{\alpha ^2} - 2$ is also a root of this equation, is

The coefficients of $x$ in the quadratic equation ${x^2} + bx + c = 0$ was taken as 17 in place of 13, its roots were found to be $- 2$ and $- 15$. The correct root of the original equation are

Let $\alpha$ and $\beta$ be the roots of equation $p{x^2} + qx + r = 0,\,\,\,p \ne 0.$ If $p,\,\,q$ and $r$ are in AP and $\frac{1}{\alpha } + \frac{1}{\beta } = 4,$ then the value of $\left| {\alpha - \beta } \right|$ is

A solution of the set of equations $x+y+z=x y+y z+z x=0$, $x^3+y^3+z^3=-3$ is-

Let $f\left( x \right)$ be a quadratic polynomial such that $f\left( { - 1} \right) + f\left( 2 \right) = 0.$ If one of the roots of $f\left( x \right) = 0$ is 3, then its other root lies in

If the roots of the equation $a{x^2} + bx + c = 0\;\,\mathrm{be}\,\;\alpha \;\mathrm{and}\;\beta \;,$ then the roots of the equation $c{x^2} + bx + a = 0$ are

The quadratic equation whose roots are cubes of the roots of $x^2+b x+c=0$ is

Value of

$x=\sqrt{6+\sqrt{6+\sqrt{6+\cdots \text { up to } \infty}}}$ (1)

is

The number of roots of the quadratic equation $8 \sec ^2 \theta-6 \sec \theta+1=0$ is:

The equation $\left( {\cos p - 1} \right){x^2} + \cos p\;x + \sin p = 0,$ in variable $x$, has real roots. Then, $p$ belongs to the interval

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

Let $f\left( x \right)$ be a quadratic polynomial such that $f\left( { - 2} \right) + f\left( 3 \right) = 0.$ If one of the roots of $f\left( x \right) = 0$ is –1, then the sum of the roots of $f\left( x \right) = 0$ is equal to:

Number of roots of the equation $x - \frac{2}{{x - 1}} = 1 - \frac{2}{{x - 1}}$ is

If $(\alpha, \beta)$ are the roots of equation $x^2-p x+q=0$ and $\left(\alpha^{\prime}, \beta^{\prime}\right)$ are that of $x^2-p^{\prime} x+q^{\prime}=0$, then $\left(\alpha-\alpha^{\prime}\right)^2+\left(\beta-\alpha^{\prime}\right)^2+\left(\alpha-\beta^{\prime}\right)^2+\left(\beta-\beta^{\prime}\right)^2=$

Let $f(x)$ be a quadratic expression positive for all real $x$. If $g(x)=f(x)-f^{\prime}(x)+f^{\prime \prime}(x)$, then for any real $x$

Two non-integer roots of the equation

$\left(x^2+3 x\right)^2-\left(x^2+3 x\right)-6=0$ (1)

$\operatorname{are}$

If one root of equation $x^2+p x+q=0$ is square of the other then :

The number of real solutions of $x^2-2 x+2+\mathrm{b}-$ $11=0$ is:

The number of real roots of the equation, ${e^{4x}} + {e^{3x}} - 4{e^{2x}} + {e^x} + 1 = 0$ is