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Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true ?

$(I)$ It has at least one negative root.

$(II)$ It has at least one positive root.

$(III)$ Both its roots are real.

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Important Questions on Theory of Equations

EASY

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The solutions of

x^{\frac{2}{5}} + 3 x^{\frac{1}{5}} - 4 = 0

are

EASY

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Let

p, q

and

r

be real numbers

(p \neq q, r \neq 0)

, such that the roots of the equation

\frac{1}{x + p} + \frac{1}{x + q} = \frac{1}{r}

are equal in magnitude but opposite in sign, then the sum of squares of these roots is equal to

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Let

r

be a root of the equation

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

. The value of

(r + 2) (r + 3) (r + 4) (r + 5)

is equal to -

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Let

f (x) = p x^{2} + q x + r

,

where

p, q, r

are constants and

p \neq 0 .

f (5) = - 3 f (2)

and

f (- 4) = 0

,

then the other root of

f

EASY

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The harmonic mean of the roots of the equation

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

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

α \neq β, α^{2} = 5 α - 3, β^{2} = 5 β - 3

,

then the equation having

\frac{α}{β}

and

\frac{β}{α}

as its roots is

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Ten ants are on the real line. At time

t = 0,

the

k

^$th$ ant starts at the point

k^{2}

and travelling at uniform speed, reaches the point

{(11 - k)}^{2}

at time

t = 1 .

The number of distinct times at which at least two ants are at the same location is

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

\frac{1}{\sqrt{α}}, \frac{1}{\sqrt{β}}

are the roots of the equation

a x^{2} + b x + 1 = 0, (a \neq 0, a, b \in R)

, then the equation

x (x + b^{3}) + (a^{3} - 3 a b x) = 0

has roots:

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

If the two roots of the equation,

(a - 1) (x^{4} + x^{2} + 1) + (a + 1) {(x^{2} + x + 1)}^{2} = 0

are real and distinct, then the set of all values of

a

is equal to

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Let

α and β

be the roots of equation

p x^{2} + q x + r = 0

p \neq 0

. If

p, q, r

are in A.P. and

\frac{1}{α} + \frac{1}{β} = 4

, then the value of

|α - β|

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

Suppose the quadratic polynomial

P (x) = a x^{2} + b x + c

has positive coefficients

a, b, c

in arithmetic progression in that order. If

P (x) = 0

has integer roots

α and β,

then

α + β + α β

equals

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

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

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The sum of all the real values of

x

Satisfying the equation

2^{(x - 1) (x^{2} + 5 x - 50)} = 1

is:

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

\sin α

and

\cos α

are the roots of the equation

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

,

then

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The number of ordered pairs

(x, y)

of real numbers that satisfy the simultaneous equations

x + y^{2} = x^{2} + y = 12

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The sum of all real values of

x

satisfying the equation

{(x^{2} - 5 x + 5)}^{x^{2} + 4 x - 60} = 1

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

The equation

\sqrt{3 x^{2} + x + 5} = x - 3

, where

x

is real, has

MEDIUM

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

For how many different values of $a$ does the following system have at least two distinct solutions?

$a x + y = 0$

$x + (a + 10) y = 0$

EASY

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

λ

be the ratio of the roots of the quadratic equation in

x, 3 m^{2} x^{2} + m (m - 4) x + 2 = 0

, then the least value of

m

for which

λ + \frac{1}{λ} = 1

, is :

HARD

Mathematics>Algebra>Theory of Equations>Solution of Quadratic Equations

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

Let a,b be non-zero real numbers. Which of the following statements about the quadratic equation ax2+a+bx+b=0 is necessarily true ? I It has at least one negative root. II It has at least one positive root. III Both its roots are real.

Important Questions on Theory of Equations

Learn and Prepare for any exam you want !

Let $a, b$ be non-zero real numbers. Which of the following statements about the quadratic equation $a x^{2} + (a + b) x + b = 0$ is necessarily true ?

$(I)$ It has at least one negative root.

$(II)$ It has at least one positive root.

$(III)$ Both its roots are real.