Quadratic Equations · Mathematics · WB JEE

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

The equation $${x^2} - 3|x| + 2 = 0$$ has

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} -...

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

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

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

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...

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

The quadratic equation $${x^2} + 15|x| + 14 = 0$$ has

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

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

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

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...

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

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 $$\mathrm{ac} \neq 0$$...

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. Th...

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

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

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

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

Let $$\alpha$$, $$\beta$$ be the roots of the equation x2 $$-$$ 6x $$-$$ 2 = 0 with $$\alpha$$ > $$\beta$$. If an = $$\alpha$$n $$-$$ $$\beta$$n for n...

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

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...

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 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 $$...

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

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

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

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

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

If the quadratic equation $$a x^2+b x+c=0(a>0)$$ has two roots $$\alpha$$ and $$\beta$$ such that $$\alpha2$$, then

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 ...

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