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If $a, b, c$ are in $AP$ and if the equations ${(b - c) x}^{2} + (c - a) x + (a - b) = 0 \dots (1)$ and $2 {(c + a) x}^{2} + (b + c) x = 0 \dots (2)$ have a common root, then

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Important Questions on Theory of Equation

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

Two distinct polynomials

f (x)

and

g (x)

are defined as follows:

f (x) = x^{2} + a x + 2; g (x) = x^{2} + 2 x + a

If the equations

f (x) = 0

and

g (x) = 0

have a common root then the sum of roots of the equation

f (x) + g (x) = 0

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

f (x) = x^{2} + a x + 2 = 0

and

g (x) = x^{2} + 2 x + a = 0

have only one real common root, then sum of the roots of

f (x) + g (x) = 0

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

The pair of equation $x + 2 y = - 5$ and $- 3 x - 6 y = - 1$ will have _____.

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

x^{2} + 2 x + 3 = 0

and

a x^{2} + b x + c = 0,

a, b, c \in R,

have a common root, then

a : b : c

is:

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

The quadratic equations

x^{2} - 6 x + a = 0

and

x^{2} - c x + 6 = 0

have one root in common. The other roots of the first and second equations are integers in the ratio

4 : 3

. Then, the common root is

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

x = 1

is a common root of the equations

a x^{2} + a x + 3 = 0

and

x^{2} + x + b = 0

, then the value of

a b

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

a

and

b

are roots of the quadratic equation

x^{2} + 5 x + d = 0

and

a

and

c

are roots of the quadratic equation

x^{2} + 6 x + 2 d = 0

. If there is only one common root of the above two equations, then the possible value of

d

is:

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

α

and

β

be the roots of equation

x^{2} - 6 x - 2 = 0

. If

a_{n} = α^{n} - β^{n}, \forall n \geq 1,

then the value of

\frac{a_{10} - 2 a_{8}}{2 a_{9}}

is equal to

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

a x^{2} + b x + c = 0

and

2 x^{2} + 3 x + 4 = 0

have a common root where

a, b, c \in N

(set of natural numbers), the least value of

a + b + c

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

The number of pairs

(a, b)

of real numbers, such that whenever

α

is a root of the equation

x^{2} + a x + b = 0, α^{2} - 2

is also a root of this equation, is :

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the quadratic equations

x^{2} - 7 x + 3 c = 0

and

x^{2} + x - 5 c = 0

have a common root, then for non-zero real value of

c

the sign of the expression

x^{2} - 3 x + c

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

x^{2} + b x - 1 = 0

and

x^{2} + x + b = 0

have a common root different from

- 1,

then

|b|

is equal to :

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the value of real number

α > 0

for which

x^{2} - 5 α x + 1 = 0

and

x^{2} - α x - 5 = 0

have a common real roots is

\frac{3}{\sqrt{2 β}}

then

β

is equal to ________

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

(x + k)

is a common factor of

(x^{2} + p x + q)

and

(x^{2} + l x + m)

, then the value of

k

is:

MEDIUM

Mathematics>Algebra>Theory of Equation>Common Roots

α

and

β

are the roots of the equation

x^{2} - 2 x + 4 = 0,

then

α^{12} + β^{12} =

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

- \frac{π}{6} < θ < - \frac{π}{12}

. Suppose

α_{1}

and

β_{1}

are the roots of the equation

x^{2} - 2 x \sec θ + 1 = 0

and

α_{2}

and

β_{2}

are the roots of the equation

x^{2} + 2 x \tan θ - 1 = 0 .

α_{1} > β_{1}

and

α_{2} > β_{2}

, then

α_{1} + β_{2}

equals

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

The harmonic mean of the roots of the equation

(5 + \sqrt{2}) x^{2} - (4 + \sqrt{5}) x + (8 + 2 \sqrt{5} = 0)

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

For

a \neq b

, if the equation

x^{2} + a x + b = 0

and

x^{2} + b x + a = 0

have a common root, then the value of

a + b =

HARD

Mathematics>Algebra>Theory of Equation>Common Roots

Let

λ \neq 0

be in

R

. If

α

and

β

are the roots of the equation,

x^{2} - x + 2 λ = 0

and

α

and

γ

are the roots of the equation,

3 x^{2} - 10 x + 27 λ = 0,

then

\frac{β γ}{λ}

is equal to:

EASY

Mathematics>Algebra>Theory of Equation>Common Roots

If the equations

x^{2} + a x + 1 = 0

and

x^{2} - x - a = 0

have a real common root

b

,

then the value of

b

is equal to

If a,b,c are in AP and if the equations (b-c)x2+c-ax+a-b=0 … 1 and 2(c+a)x2+b+cx=0 … 2 have a common root, then

Important Questions on Theory of Equation

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If $a, b, c$ are in $AP$ and if the equations ${(b - c) x}^{2} + (c - a) x + (a - b) = 0 \dots (1)$ and $2 {(c + a) x}^{2} + (b + c) x = 0 \dots (2)$ have a common root, then