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If equations $x^{6} + a x^{4} - 4 x = 0$ and $x^{7} + a x^{5} - 1 = 0$ have a common root, then identify which of the following statement is(are) correct?

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Important Questions on Quadratic Equations

MEDIUM

Mathematics>Algebra>Quadratic Equations>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

EASY

Mathematics>Algebra>Quadratic Equations>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>Quadratic Equations>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:

MEDIUM

Mathematics>Algebra>Quadratic Equations>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>Quadratic Equations>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>Quadratic Equations>Common Roots

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

HARD

Mathematics>Algebra>Quadratic Equations>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>Quadratic Equations>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:

HARD

Mathematics>Algebra>Quadratic Equations>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>Quadratic Equations>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

EASY

Mathematics>Algebra>Quadratic Equations>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

EASY

Mathematics>Algebra>Quadratic Equations>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

EASY

Mathematics>Algebra>Quadratic Equations>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 ________

MEDIUM

Mathematics>Algebra>Quadratic Equations>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 :

HARD

Mathematics>Algebra>Quadratic Equations>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>Quadratic Equations>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>Quadratic Equations>Common Roots

If the quadratic equations

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

and

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

have a common root then

a + 4 b + 4 c

is equal to

EASY

Mathematics>Algebra>Quadratic Equations>Common Roots

If quadratic equation

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

and

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

and

b \neq c

have a common root then

b + c - 3

is equal to

HARD

Mathematics>Algebra>Quadratic Equations>Common Roots

The value of

a

for which the equations

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

and

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

have a common root is -

HARD

Mathematics>Algebra>Quadratic Equations>Common Roots

The values of

k

for which the equations

x^{2} - k x - 21 = 0

and

x^{2} - 3 k x + 35 = 0

will have a common root are

If equations x6+ax4-4x=0 and x7+ax5-1=0 have a common root, then identify which of the following statement is(are) correct?

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If equations $x^{6} + a x^{4} - 4 x = 0$ and $x^{7} + a x^{5} - 1 = 0$ have a common root, then identify which of the following statement is(are) correct?