If the equation a x 2 + b x + c = 0 and b x 2 + c x + a = 0 (where a c ≠ b 2 ) have a common root, then show that, either a + b + c = 0 or a = b = c .

let α is the common roots of ax 2 + bx + c = 0 and bx 2 + cx + a = 0 aα 2 + bα + c = 0 and bα 2 + cα + a = 0 Solving by the rule of cross multiplication,we get α 2 a b - c 2 = α c b - a 2 = 1 a c - b 2 ⇒ α or = a b - c 2 c b - a 2 c b - a 2 a c - b 2 ∴ a b - c 2 c b - a 2 = c b - a 2 a c - b 2 ⇒ a a 3 + b 3 + c 3 - 3 a b c = 0 ⇒ a = 0 or a 3 + b 3 + c 3 - 3 a b c = 0 ⇒ a + b + c a 2 + b 2 + c 2 - a b - b c - c a = 0 a + b + c 1 2 a - b 2 + b - c 2 + c - a 2 = 0 ⇒ a + b + c = 0 or, ⇒ a - b 2 + b - c 2 + c - a 2 = 0 The second equation possible only when, a - b = 0 , b - c = 0 , c - a = 0 ⇒ a = b = c answer. ,

If the equation a x 2 + b x + c = 0 and b x 2 + c x + a = 0 (where a c ≠ b 2 ) have a common root, then show that, either a + b + c = 0 or a = b = c .

let α is the common roots of ax 2 + bx + c = 0 and bx 2 + cx + a = 0 aα 2 + bα + c = 0 and bα 2 + cα + a = 0 Solving by the rule of cross multiplication,we get α 2 a b - c 2 = α c b - a 2 = 1 a c - b 2 ⇒ α or = a b - c 2 c b - a 2 c b - a 2 a c - b 2 ∴ a b - c 2 c b - a 2 = c b - a 2 a c - b 2 ⇒ a a 3 + b 3 + c 3 - 3 a b c = 0 ⇒ a = 0 or a 3 + b 3 + c 3 - 3 a b c = 0 ⇒ a + b + c a 2 + b 2 + c 2 - a b - b c - c a = 0 a + b + c 1 2 a - b 2 + b - c 2 + c - a 2 = 0 ⇒ a + b + c = 0 or, ⇒ a - b 2 + b - c 2 + c - a 2 = 0 The second equation possible only when, a - b = 0 , b - c = 0 , c - a = 0 ⇒ a = b = c answer. ,

Let α and β be the roots of x 2 - x - 1 = 0 , with α > β . For all positive integers n , define a n = α n - β n α - β , n ≥ 1 b 1 = 1 and b n = a n - 1 + a n + 1 , n ≥ 2 . Then which of the following options is/are correct?

b n = α n + β n for all n ≥ 1

MEDIUM

Earn 100

If the equation $a x^{2} + b x + c = 0$ and $b x^{2} + c x + a = 0$ (where $a c \neq b^{2}$ ) have a common root, then show that, either $a + b + c = 0$ or $a = b = c$ .

Important Questions on Quadratic Equations

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

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>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 =

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

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

(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

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

HARD

Mathematics>Algebra>Quadratic Equations>Common Roots

Let

S

be the set of all non-zero real numbers

α

such that the quadratic equation

α x^{2} - x + α = 0

has two distinct real roots

x_{1} & x_{2}

satisfying the inequality

|x_{1} - x_{2}| < 1 .

Which of the following intervals is(are) a subset(s) of

S

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:

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

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

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

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>Quadratic Equations>Common Roots

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

MEDIUM

Mathematics>Algebra>Quadratic Equations>Common Roots

α

and

β

are the roots of the equation

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

then

α^{12} + β^{12} =

MEDIUM

Mathematics>Algebra>Quadratic Equations>Common Roots

If the equation

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

has roots equal to the fourth powers of the roots of

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

where

a^{2} > 4 b,

then the roots of

x^{2} - 4 b x + 2 b^{2} - c = 0

will be

HARD

Mathematics>Algebra>Quadratic Equations>Common Roots

Let

α

and

β

be the roots of

x^{2} - x - 1 = 0,

with

α > β .

For all positive integers

n,

define

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

b_{1} = 1

and

b_{n} = a_{n - 1} + a_{n + 1}, n \geq 2 .

Then which of the following options is/are correct?

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 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 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:

If the equation ax2+bx+c=0 and bx2+cx+a=0 (where ac≠b2) have a common root, then show that, either a+b+c=0 or a=b=c.

Important Questions on Quadratic Equations

Learn and Prepare for any exam you want !

If the equation $a x^{2} + b x + c = 0$ and $b x^{2} + c x + a = 0$ (where $a c \neq b^{2}$ ) have a common root, then show that, either $a + b + c = 0$ or $a = b = c$ .