Quadratic Equations

If $\alpha, \beta$ are the roots of the equation $x^2-p x+q=0$ and $\alpha>0, \beta>0$, then $\alpha^{\frac{1}{4}}+\beta^{\frac{1}{4}}=\left(p+6 \sqrt{p}+4 q^{\frac{1}{4}} \sqrt{p+2 \sqrt{q}}\right)^k$, where $K$ is

If $0<\alpha<\beta<\gamma<\frac{\pi}{2}$, then the equation $\frac{1}{x-\sin \alpha}+\frac{1}{x-\sin \beta}+\frac{1}{x-\sin \gamma}=0$ has

If $\left(4^{\sec ^2 \alpha}\right) x^2+2 x+\left(\beta^2-\beta+\frac{1}{2}\right)=0$ has real roots，then the value／values of $\left(\cos \alpha+\cos ^{-1} \beta\right)$ is／are

The equation $x^3+5 x^2+p x+q=0$ and $x^3+7 x^2+p x+r=0$ have two roots in common. If the third root of each equation is represented by $x_1$ and $x_2$ respectively, the GCD of $x_1, x_2$ will be

Let 1 lies between the roots of the equation $y^2-m y+1=0$ and $[x]$ denotes the greatest integer function. Then the value of $\left[\left(\frac{4|x|}{x^2+16}\right)^m\right]$ is

For what value of ' $a$ ', the sum of the squares of the roots of the equation $x^2-(a-2) x-a+1=0$ will have the least value?

If the sum of the squares of the roots of the equation $x^2-(a-2) x-(a+1)=0$ is least for an appropriate value of the variable parameter $a$, then that value of ' $a$ ' will be

If the quadratic equation \(a x^2+b x+c=0(a>0)\) has two roots \(\alpha\) and \(\beta\) such that \(\alpha<-2\) and \(\beta>2\), then

If \(a, b, c\) are distinct odd natural numbers, then the number of rational roots of the equation \(a x^2+b x+c=0\)

Let \(\mathrm{N}\) be the number of quadratic equations with coefficients from \(\{0,1,2, \ldots, 9\}\) such that 0 is a solution of each equation. Then the value of \(\mathrm{N}\) is

If \(\mathrm{P}(x)=\mathrm{a} x^2+\mathrm{b} x+\mathrm{c}\) and \(\mathrm{Q}(x)=-\mathrm{a} x^2+\mathrm{d} x+\mathrm{c}\) where \(\mathrm{ac} \neq 0\), then \(\mathrm{P}(x) \cdot \mathrm{Q}(x)=0\) has (a, b, c, d are real)

If one root of \({x^2} + px - {q^2} = 0,p\) and \(q\) are real, be less than 2 and other be greater than 2, then

The value of a for which the sum of the squares of the roots of the equation \({x^2} - (a - 2)x - a - 1 = 0\) assumes the least value is

If a, b are odd integers, then the roots of the equation \(2a{x^2} + (2a + b)x + b = 0\), \(a \ne 0\) are

Let \(\alpha\), \(\beta\) be the roots of the equation x2 \(-\) 6x \(-\) 2 = 0 with \(\alpha\) > \(\beta\). If an = \(\alpha\)n \(-\) \(\beta\)n for n \(\ge\) 1, then the value of \({{{a_{10}} - 2{a_8}} \over {2{a_9}}}\) is

If P(x) = ax2 + bx + c and Q(x) = \(-\)ax2 + dx + c, where ac \(\ne\) 0 [a, b, c, d are all real], then P(x).Q(x) = 0 has

Let z1 and z2 be two imaginary roots of z2 + pz + q = 0, where p and q are real. The points z1, z2 and origin form an equilateral triangle if

The expression ax2 + bx + c (a, b and c are real) has the same sign as that of a for all x if

Let a, b, c be real numbers such that a + b + c < 0 and the quadratic equation ax2 + bx + c = 0 has imaginary roots. Then,

If \({b_1}{b_2} = 2({c_1} + {c_2})\) and b1, b2, c1, c2 are all real numbers, 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

If the equation \({x^2} - cx + d = 0\) has roots equal to the fourth powers of the roots of \({x^2} + ax + b = 0\), where \({a^2} > 4b\), then the roots of \({x^2} - 4bx + 2{b^2} - c = 0\) will be

If a, b\(\in\) {1, 2, 3} and the equation ax2 + bx + 1 = 0 has real roots, then

If p, q are odd integers, then the roots of the equation \(2p{x^2} + (2p + q)x + q = 0\) are

Let \(\alpha\) and \(\beta\) be the roots of \({x^2} + x + 1 = 0\). If n be a positive integer, then \(\alpha\)n + \(\beta\)n is

The greatest integer which divides \((p + 1)(p + 2)(p + 3)...(p + q)\) for all \(p \in N\) and fixed \(q \in N\) is

For real x, the greatest value of \({{{x^2} + 2x + 4} \over {2{x^2} + 4x + 9}}\) is

The number of values of k, for which the equation x2 \(-\) 3x + k = 0 has two distinct roots lying in the interval (0, 1), are

If \(\alpha\) and \(\beta\) are roots of ax2 + bx + c = 0, then the equation whose roots are \(\alpha\)2 and \(\beta\)2, is

If p, q are the roots of the equation x2 + px + q = 0, then

If the equation x2 + y2 \(-\) 10x + 21 = 0 has real roots x = \(\alpha\) and y = \(\beta\), then

The quadratic equation 2x2 \(-\) (a3 + 8a \(-\) 1)x + a2 \(-\) 4a = 0 possesses roots of opposite sign. Then,

Determine the sum of imaginary roots of the equation \((2{x^2} + x - 1)(4{x^2} + 2x - 3) = 6\)

If \(\alpha\) and \(\beta\) are the roots of the equation x2 + x + 1 = 0, then the equation whose roots ar \(\alpha\)19 and \(\beta\)7 is

If the ratio of the roots of the equation \(p{x^2} + qx + r = 0\) is \(a:b\), then \({{ab} \over {{{(a + b)}^2}}}\) =

Let a, b, c be three real numbers such that a + 2b + 4c = 0. Then the equation ax2 + bx + c = 0

If sin \(\theta\) and cos \(\theta\) are the roots of equation ax2 \(-\) bx + c = 0, then a, b and c satisfy the relation

The quadratic equation \({x^2} + 15|x| + 14 = 0\) has

The roots of the quadratic equation \({x^2} - 2\sqrt 3 x - 22 = 0\) are

If \(\alpha\), \(\beta\) be the roots of the quadratic equation x2 + x + 1 = 0 then the equation whose roots are \(\alpha\)19, \(\beta\)7 is

Find the values of a for which the expression \({x^2} - (3a - 1)x + 2{a^2} + 2a - 11\) is always positive.

If a, b, c are real, then both the roots of the equation \((x - b)(x - c) + (x - c)(x - a) + (x - a)(x - b) = 0\) are always

The quadratic equation whose roots are three times the roots of \(3a{x^2} + 3bx + c = 0\) is

The sum of all real roots of the equation \(|x - 2{|^2} + |x - 2| - 2 = 0\) is

If \(\alpha\), \(\beta\) be the roots of \({x^2} - a(x - 1) + b = 0\), then the value of \({1 \over {{\alpha ^2} - a\alpha }} + {1 \over {{\beta ^2} - a\beta }} + {2 \over {a + b}}\) is

The equation \({x^2} - 3|x| + 2 = 0\) has

If one root of the equation \({x^2} + (1 - 3i)x - 2(1 + i) = 0\) is \(-\)1 + i, then the other root is