If rs are the roots of a0a1xa2x20a0a1a2Ra20 then the equality a0a1xa2x2a01xr1xsholds

EASY

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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

$a x + y = 0$

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

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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.

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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|

is

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

Let

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

,

where

p, q, r

are constants and

p \neq 0 .

If

f (5) = - 3 f (2)

and

f (- 4) = 0

,

then the other root of

f

is

EASY

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

The harmonic mean of the roots of the equation

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

is

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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 .

If

α_{1} > β_{1}

and

α_{2} > β_{2}

, then

α_{1} + β_{2}

equals

EASY

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

The solutions of

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

are

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

Difference between the corresponding roots of

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

is same and

a \neq b

,⁡ then

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

If

\sin α

and

\cos α

are the roots of the equation

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

,

then

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

If

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

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

The sum of all the real values of

x

Satisfying the equation

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

is:

HARD

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

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

EASY

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

If

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 :

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

If

\tan A

and

\tan B

are the roots of the quadratic equation

3 x^{2} - 10 x - 25 = 0

, then the value of

3 \sin^{2} (A + B) - 10 \sin (A + B) \cos (A + B) - 25 \cos^{2} (A + B)

is :

MEDIUM

Mathematics>Algebra>Reasoning with Equations and Inequalities>Solve quadratic equations in one variable. a. Use the method of completing the square to transform any quadratic equation in x into an equation of the form (x – p)2 = q that has the same solutions. Derive the quadratic formula from this form. b. Solve quadratic equations by inspection, taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b

The sum of all the real roots of the equation

{|x - 2|}^{2} + |x - 2| - 2 = 0

is

If $r, s$ are the roots of $a_{0} + a_{1} x + a_{2} x^{2} = 0 (a_{0}, a_{1}, a_{2} \in R & a_{2} \neq 0)$ , then the equality $a_{0} + a_{1} x + a_{2} x^{2} = a_{0} (1 - \frac{x}{r}) (1 - \frac{x}{s})$ holds

Important Questions on Reasoning with Equations and Inequalities

Learn and Prepare for any exam you want !

If r, s are the roots of a0+a1x+a2x2=0 a0, a1, a2∈R & a2≠0, then the equality a0+a1x+a2x2=a01-xr1-xs holds

Important Questions on Reasoning with Equations and Inequalities

Learn and Prepare for any exam you want !

If $r, s$ are the roots of $a_{0} + a_{1} x + a_{2} x^{2} = 0 (a_{0}, a_{1}, a_{2} \in R & a_{2} \neq 0)$ , then the equality $a_{0} + a_{1} x + a_{2} x^{2} = a_{0} (1 - \frac{x}{r}) (1 - \frac{x}{s})$ holds