If α, β are the roots of x² + px + q =...

MathematicsCommon RootsJEE Advanced 1979Moderate

If $α, β$ are the roots of $x^{2} + p x + q = 0$ and $γ, δ$ are the roots of $x^{2} + r x + s = 0$ , evaluate $(α - γ) (α - δ) (β - γ) (β - δ)$ in terms of $p, q, r$ and $s$ . Deduce the condition that the equations have a common root.

Answer:

(q - s)^{2} - (p - r) (p s - q r); Condition: (q - s)^{2} - (p - r) (p s - q r) = 0

Visualized Solution (English)

Defining the Roots and Equations

Given Equation 1: $x^{2} + p x + q = 0$ with roots $α, β$ .
Given Equation 2: $x^{2} + r x + s = 0$ with roots $γ, δ$ .
We need to evaluate: $E = (α - γ) (α - δ) (β - γ) (β - δ)$ .

The Polynomial Identity Trick

Since $γ, δ$ are roots of $x^{2} + r x + s = 0$ :
$(x - γ) (x - δ) = x^{2} + r x + s$
Substituting $x = α$ :
$(α - γ) (α - δ) = α^{2} + r α + s$

Simplifying using the First Equation

Since $α$ is a root of $x^{2} + p x + q = 0$ :
$α^{2} + p α + q = 0 ⟹ α^{2} = - p α - q$
Substitute $α^{2}$ into the expression from Step 1:
$(α - γ) (α - δ) = (- p α - q) + r α + s$
$= (r - p) α + (s - q)$

Symmetry for the Beta Terms

Similarly, for the root $β$ :
$(β - γ) (β - δ) = (r - p) β + (s - q)$
The total product $E$ becomes:
$E = [(r - p) α + (s - q)] \times [(r - p) β + (s - q)]$

Expanding the Product

Expanding the product:
$E = (r - p)^{2} (α β) + (r - p) (s - q) (α + β) + (s - q)^{2}$

Substituting Sum and Product of Roots

From $x^{2} + p x + q = 0$ , we have $α + β = - p$ and $α β = q$ .
Substitute these into $E$ :
$E = (r - p)^{2} (q) + (r - p) (s - q) (- p) + (s - q)^{2}$
$E = q (r - p)^{2} - p (r - p) (s - q) + (s - q)^{2}$

Final Algebraic Simplification

Rearranging the terms:
$E = (q - s)^{2} + (r - p) [q (r - p) - p (s - q)]$
$E = (q - s)^{2} + (r - p) [q r - q p - p s + pq]$
$E = (q - s)^{2} + (r - p) (q r - p s)$
Final Form: $E = (q - s)^{2} - (p - r) (p s - q r)$

Condition for a Common Root

Condition for Common Root:
If the equations have a common root, then at least one factor in $E$ must be zero.
Therefore, $E = 0$ .
Condition: $(q - s)^{2} - (p - r) (p s - q r) = 0$

00:00 / 00:00

Conceptually Similar Problems

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2007Easy

View in:English Hindi

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

(A)

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

(B)

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

(C)

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

(D)

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

Answer:D

MathematicsNature of RootsJEE Advanced 1989Easy

View in:English Hindi

If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $α^{4}, β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the equation $x^{2} - 4 q x + 2 q^{2} - r = 0$ has always

(A)

two real roots

(B)

two positive roots

(C)

two negative roots

(D)

one positive and one negative root

Answer:A

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2004Easy

View in:English Hindi

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

(A)

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

(B)

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

(C)

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

(D)

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

Answer:A

MathematicsLocation of RootsJEE Advanced 1989Moderate

View in:English Hindi

Let $a, b, c$ be real numbers, $a \neq = 0$ . If $α$ is a root of $a^{2} x^{2} + b x + c = 0$ . $β$ is the root of $a^{2} x^{2} - b x - c = 0$ and $0 < α < β$ , then the equation $a^{2} x^{2} + 2 b x + 2 c = 0$ has a root $γ$ that always satisfies

(A)

γ = \frac{α + β}{2}

(B)

γ = α + \frac{β}{2}

(C)

γ = α

(D)

α < γ < β

Answer:D

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2010Easy

View in:English Hindi

Let $p$ and $q$ be real numbers such that $p \neq = 0, p^{3} \neq = - q$ . If $α$ and $β$ are nonzero complex numbers satisfying $α + β = - p$ and $α^{3} + β^{3} = q$ , then a quadratic equation having $α / β$ and $β / α$ as its roots is

(A)

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

(B)

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

(C)

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

(D)

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

Answer:B

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1992Easy

View in:English Hindi

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

(A)

a, c

(B)

b, c

(C)

a, b

(D)

a + c, b + c

Answer:C

MathematicsCommon RootsJEE Advanced 1986Easy

View in:English Hindi

If the quadratic equations $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ ( $a \neq = b$ ) have a common root, then the numerical value of $a + b$ is .........

Answer:-1

MathematicsRelation Between Roots and CoefficientsJEE Main 2002Easy

View in:English Hindi

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

(A)

a + b + 4 = 0

(B)

a + b - 4 = 0

(C)

a - b - 4 = 0

(D)

a - b + 4 = 0

Answer:A

MathematicsCommon RootsJEE Main 2009Easy

View in:English Hindi

The quadritic 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

(A)

(B)

(C)

(D)

Answer:D

MathematicsCommon RootsJEE Advanced 1985Moderate

View in:English Hindi

If $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

(A)

A.P.

(B)

G.P.

(C)

H.P.

(D)

none of these

Answer:A

Visualized Solution (English)

Defining the Roots and Equations

The Polynomial Identity Trick

Simplifying using the First Equation

Symmetry for the Beta Terms

Expanding the Product

Substituting Sum and Product of Roots

Final Algebraic Simplification

Condition for a Common Root

Conceptually Similar Problems

.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; } If α and β are the roots of x2+px+q=0 and α4,β4 are the roots of x2−rx+s=0, then the equation x2−4qx+2q2−r=0 has always

.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 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 the quadratic equations x2+ax+b=0 and x2+bx+a=0 (a=b) have a common root, then the numerical value of a+b 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 a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in

.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; } If α and β are the roots of x2+px+q=0 and α4,β4 are the roots of x2−rx+s=0, then the equation x2−4qx+2q2−r=0 has always

.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 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 the quadratic equations x2+ax+b=0 and x2+bx+a=0 (a=b) have a common root, then the numerical value of a+b 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 a,b,c are in G.P., then the equations ax2+2bx+c=0 and dx2+2ex+f=0 have a common root if d/a,e/b,f/c are in

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

If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $α^{4}, β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the equation $x^{2} - 4 q x + 2 q^{2} - r = 0$ has always

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

If the quadratic equations $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ ( $a \neq = b$ ) have a common root, then the numerical value of $a + b$ 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 $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in

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

If $α$ and $β$ are the roots of $x^{2} + p x + q = 0$ and $α^{4}, β^{4}$ are the roots of $x^{2} - r x + s = 0$ , then the equation $x^{2} - 4 q x + 2 q^{2} - r = 0$ has always

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

If the quadratic equations $x^{2} + a x + b = 0$ and $x^{2} + b x + a = 0$ ( $a \neq = b$ ) have a common root, then the numerical value of $a + b$ 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 $a, b, c$ are in G.P., then the equations $a x^{2} + 2 b x + c = 0$ and $d x^{2} + 2 e x + f = 0$ have a common root if $d / a, e / b, f / c$ are in