Relation Between Roots and Coefficients

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- 3

- 6

34 /9

61 /9

(3, \infty)

(- \infty, - 3)

(- 3, 3)

(- 3, \infty)

1

0

3

2

- 2

3

2

1

x^{2} - 18 x - 16 = 0

x^{2} - 18 x + 16 = 0

x^{2} + 18 x - 16 = 0

x^{2} + 18 x + 16 = 0

- 1/3

- 2/3

3 x^{2} - 19 x + 3 = 0

3 x^{2} + 19 x - 3 = 0

3 x^{2} - 19 x - 3 = 0

x^{2} - 5 x + 3 = 0

a + b + 4 = 0

a + b - 4 = 0

a - b - 4 = 0

a - b + 4 = 0

p = 1, q = - 2

p = 0, q = 1

p = - 2, q = 0

p = - 2, q = 1

2 (sec θ - tan θ)

2 sec θ

- 2 tan θ

0

(p^{3} + q) x^{2} - (p^{3} + 2 q) x + (p^{3} + q) = 0

(p^{3} + q) x^{2} - (p^{3} - 2 q) x + (p^{3} + q) = 0

(p^{3} - q) x^{2} - (5 p^{3} - 2 q) x + (p^{3} - q) = 0

(p^{3} - q) x^{2} - (5 p^{3} + 2 q) x + (p^{3} - q) = 0

\frac{2}{9} (p - q) (2 q - p)

\frac{2}{9} (q - p) (2 p - q)

\frac{2}{9} (q - 2 p) (2 q - p)

\frac{2}{9} (2 p - q) (2 q - p)

Δ \neq = 0

b Δ = 0

c Δ = 0

Δ = 0

p^{3} - q (3 p - 1) + q^{2} = 0

p^{3} - q (3 p + 1) + q^{2} = 0

p^{3} + q (3 p - 1) + q^{2} = 0

p^{3} + q (3 p + 1) + q^{2} = 0

0 < α < β

α < 0 < β < ∣ α ∣

α < β < 0

α < 0 < ∣ α ∣ < β

a, c

b, c

a, b

a + c, b + c

a^{2} + c^{2} = - ab

a^{2} - c^{2} = - ab

a^{2} - c^{2} = ab

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be the roots of equation x2−6x−2=0. If an​=αn−βn, for n≥1, then the value of 2a9​a10​−2a8​​ is equal to :

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be the roots of equation px2+qx+r=0,p=0. If p,q,r are in A.P. and 1/α+1/β=4, then the value of ∣α−β∣ is:

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the difference between the roots of the equation x2+ax+1=0 is less than 5​, then the set of possible values of a is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of a for which the sum of the squares of the roots of the equation x2−(a−2)x−a−1=0 assume the least value is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the roots of the equation x2−bx+c=0 be two consecutive integers, then b2−4c equals

Let two numbers have arithmetic mean 9 and geometric mean 4. Then these numbers are the roots of the quadratic equation

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The value of 'a' for which one root of the quadratic equation (a2−5a+3)x2+(3a−1)x+2=0 is twice as large as the other is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the sum of the roots of the quadratic equation ax2+bx+c=0 is equal to the sum of the squares of their reciprocals, then a/c,b/a and c/b are in

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α=β but α2=5α−3 and β2=5β−3 then the equation having α/β and β/α as its roots is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Difference between the corresponding roots of x2+ax+b=0 and x2+bx+a=0 is same and a=b, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If p and q are the roots of the equation x2+px+q=0, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α and β be the roots of x2−6x−2=0, with α>β. If an​=αn−βn for n≥1, then the value of 2a9​a10​−2a8​​ is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of the equation x2−px+r=0 and α/2,2β be the roots of the equation x2−qx+r=0. Then the value of r is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } In the quadratic equation ax2+bx+c=0, Δ=b2−4ac and α+β,α2+β2,α3+β3, are in G.P. where α,β are the root of ax2+bx+c=0, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root is square of the other root of the equation x2+px+q=0, then the relation between p and q is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of x2−x+p=0 and γ,δ be the roots of x2−4x+q=0. If α,β,γ,δ are in G.P., then the integral values of p and q respectively, are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α and β (α<β) are the roots of the equation x2+bx+c=0, where c<0<b, then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } For the equation 3x2+px+3=0,p>0, if one of the root is square of the other, then p is equal to

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } The harmonic mean of the roots of the equation (5+2​)x2−(4+5​)x+8+25​=0 is

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let α,β be the roots of the equation (x−a)(x−b)=c,c=0. Then the roots of the equation (x−α)(x−β)+c=0 are

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If (x2+px+1) is a factor of (ax3+bx+c), then

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a and b be the roots of the equation x2−10cx−11d=0 and those of x2−10ax−11b=0 are c,d then value of a+b+c+d, when a=b=c=d, is .........

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If the product of the roots of the equation x2−3kx+2e2lnk−1=0 is 7, then the roots are real for k=…

Let $α$ and $β$ be the roots of equation $x^{2} - 6 x - 2 = 0$ . If $a_{n} = α^{n} - β^{n}$ , for $n \geq 1$ , then the value of $\frac{a _{10} - 2 a _{8}}{2 a _{9}}$ is equal to :

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 $1/ α + 1/ β = 4$ , then the value of $∣ α - β ∣$ is:

If the difference between the roots of the equation $x^{2} + a x + 1 = 0$ is less than $5$ , then the set of possible values of $a$ is

The value of $a$ for which the sum of the squares of the roots of the equation $x^{2} - (a - 2) x - a - 1 = 0$ assume the least value is

If the roots of the equation $x^{2} - b x + c = 0$ be two consecutive integers, then $b^{2} - 4 c$ equals

The value of 'a' for which one root of the quadratic equation $(a^{2} - 5 a + 3) x^{2} + (3 a - 1) x + 2 = 0$ is twice as large as the other is

If the sum of the roots of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the sum of the squares of their reciprocals, then $a / c, b / a$ and $c / b$ are in

If $α \neq = β$ but $α^{2} = 5 α - 3$ and $β^{2} = 5 β - 3$ then the equation having $α / β$ and $β / α$ as its roots is

Difference between the corresponding roots of $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ is same and $a \neq = b$ , then

If $p$ and $q$ are the roots of the equation $x^{2} + p x + q = 0$ , then

Let $α$ and $β$ be the roots of $x^{2} - 6 x - 2 = 0$ , with $α > β$ . If $a_{n} = α^{n} - β^{n}$ for $n \geq 1$ , then the value of $\frac{a _{10} - 2 a _{8}}{2 a _{9}}$ is

Let $α, β$ be the roots of the equation $x^{2} - p x + r = 0$ and $α /2, 2 β$ be the roots of the equation $x^{2} - q x + r = 0$ . Then the value of $r$ is

In the quadratic equation $a x^{2} + b x + c = 0$ , $Δ = b^{2} - 4 a c$ and $α + β, α^{2} + β^{2}, α^{3} + β^{3}$ , are in G.P. where $α, β$ are the root of $a x^{2} + b x + c = 0$ , then

If one root is square of the other root of the equation $x^{2} + p x + q = 0$ , then the relation between $p$ and $q$ is

Let $α, β$ be the roots of $x^{2} - x + p = 0$ and $γ, δ$ be the roots of $x^{2} - 4 x + q = 0$ . If $α, β, γ, δ$ are in G.P., then the integral values of $p$ and $q$ respectively, are

If $α$ and $β$ ( $α < β$ ) are the roots of the equation $x^{2} + b x + c = 0$ , where $c < 0 < b$ , then

For the equation $3 x^{2} + p x + 3 = 0, p > 0$ , if one of the root is square of the other, then $p$ is equal to

The harmonic mean of the roots of the equation $(5 + 2) x^{2} - (4 + 5) x + 8 + 25 = 0$ is

Let $α, β$ be the roots of the equation $(x - a) (x - b) = c, c \neq = 0$ . Then the roots of the equation $(x - α) (x - β) + c = 0$ are

If $(x^{2} + p x + 1)$ is a factor of $(a x^{3} + b x + c)$ , then

Let $a$ and $b$ be the roots of the equation $x^{2} - 10 c x - 11 d = 0$ and those of $x^{2} - 10 a x - 11 b = 0$ are $c, d$ then value of $a + b + c + d$ , when $a \neq = b \neq = c \neq = d$ , is .........

If the product of the roots of the equation $x^{2} - 3 k x + 2 e^{2 l n k} - 1 = 0$ is $7$ , then the roots are real for $k = \dots$