If α, β are the roots of ax² + bx + c =...

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2000Easy

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

Visualized Solution (English)

Visualizing the Root Shift

Consider the first quadratic equation $a x^{2} + b x + c = 0$ with roots $α, β$ .
Consider the second quadratic equation $A x^{2} + B x + C = 0$ with roots $α + δ, β + δ$ .
Observe that the distance between roots is invariant under a horizontal shift by $δ$ .

Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } a x^{2} + b x + c = 0

For $a x^{2} + b x + c = 0$ , roots are $α$ and $β$ .
Sum of roots: $α + β = - \frac{b}{a}$
Product of roots: $α β = \frac{c}{a}$
Square of difference: $(α - β)^{2} = (α + β)^{2} - 4 α β$

Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } A x^{2} + B x + C = 0

For $A x^{2} + B x + C = 0$ , roots are $α^{'} = α + δ$ and $β^{'} = β + δ$ .
Difference of roots: $α^{'} - β^{'} = (α + δ) - (β + δ) = α - β$
Square of difference: $(α^{'} - β^{'})^{2} = (α - β)^{2}$

Equating Root Differences

Since the difference between roots is equal: $(α - β)^{2} = ((α + δ) - (β + δ))^{2}$
This implies that the spread of the roots is identical for both parabolas.

Discriminant Substitution

Recall the identity: $(α - β)^{2} = \frac{D}{a ^{2}} = \frac{b ^{2} - 4 a c}{a ^{2}}$
Substitute for the first equation: $\frac{b ^{2} - 4 a c}{a ^{2}}$
Substitute for the second equation: $\frac{B ^{2} - 4 A C}{A ^{2}}$
Equating both sides: $\frac{b ^{2} - 4 a c}{a ^{2}} = \frac{B ^{2} - 4 A C}{A ^{2}}$

Final Invariant Form

Key Takeaway: The ratio $\frac{D}{a ^{2}}$ is invariant under horizontal translation of the roots.
Next Challenge: What happens to this ratio if the roots are scaled by a factor $k$ instead of being shifted?
Final Result: $\frac{b ^{2} - 4 a c}{a ^{2}} = \frac{B ^{2} - 4 A C}{A ^{2}}$

00:00 / 00:00

Conceptually Similar Problems

MathematicsRelation Between Roots and CoefficientsJEE Advanced 1983Easy

View in:English Hindi

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

MathematicsLocation of RootsJEE Advanced 1995Easy

View in:English Hindi

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

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 2001Easy

View in:English Hindi

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

Answer:

α^{2} β, α β^{2}

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

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

MathematicsRelation Between Roots and CoefficientsJEE Advanced 2000Easy

View in:English Hindi

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

(A)

0 < α < β

(B)

α < 0 < β < ∣ α ∣

(C)

α < β < 0

(D)

α < 0 < ∣ α ∣ < β

Answer:B

MathematicsTrigonometric Functions of Compound AnglesJEE Main 2005Moderate

View in:English Hindi

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

(A)

a = b + c

(B)

c = a + b

(C)

b = c

(D)

b = a + c

Answer:B

MathematicsRelation Between Roots and CoefficientsJEE Main 2005Easy

View in:English Hindi

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

(A)

- 2

(B)

3

(C)

2

(D)

1

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

Visualized Solution (English)

Visualizing the Root Shift

Roots of $a x^{2} + b x + c = 0$

Roots of $A x^{2} + B x + C = 0$

Equating Root Differences

Discriminant Substitution

Final Invariant Form

Conceptually Similar Problems

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

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

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

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

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

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

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

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

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

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

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

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

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

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

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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

Visualized Solution (English)

Visualizing the Root Shift

Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } ax2+bx+c=0

Roots of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Ax2+Bx+C=0

Equating Root Differences

Discriminant Substitution

Final Invariant Form

Conceptually Similar Problems

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the quadratic equation ax2+bx+c=0 is equal to the n-th power of the other, then show that (acn)1/(n+1)+(anc)1/(n+1)+b=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a=0 and let α,β be the roots of the equation ax2+bx+c=0. Express the roots of a3x2+abcx+c3=0 in terms of α,β.

.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; } 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 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; } In a triangle PQR,∠R=π/2. If tan(P/2) and tan(Q/2) are the roots of ax2+bx+c=0,a=0 then

.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

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If α,β are the roots of ax2+bx+c=0,(a=0) and α+δ,β+δ are the roots of Ax2+Bx+C=0,(A=0) for some constant δ, then prove that (b2−4ac)/a2=(B2−4AC)/A2

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } If one root of the quadratic equation ax2+bx+c=0 is equal to the n-th power of the other, then show that (acn)1/(n+1)+(anc)1/(n+1)+b=0

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real. If ax2+bx+c=0 has two real roots α and β, where α<−1 and β>1, then show that 1+c/a+∣b/a∣<0.

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let a,b,c be real numbers with a=0 and let α,β be the roots of the equation ax2+bx+c=0. Express the roots of a3x2+abcx+c3=0 in terms of α,β.

.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; } 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 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; } In a triangle PQR,∠R=π/2. If tan(P/2) and tan(Q/2) are the roots of ax2+bx+c=0,a=0 then

.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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

Roots of $a x^{2} + b x + c = 0$

Roots of $A x^{2} + B x + C = 0$

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

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

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

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

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

If $α, β$ are the roots of $a x^{2} + b x + c = 0, (a \neq = 0)$ and $α + δ, β + δ$ are the roots of $A x^{2} + B x + C = 0, (A \neq = 0)$ for some constant $δ$ , then prove that $(b^{2} - 4 a c) / a^{2} = (B^{2} - 4 A C) / A^{2}$

If one root of the quadratic equation $a x^{2} + b x + c = 0$ is equal to the $n$ -th power of the other, then show that $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

Let $a, b, c$ be real. If $a x^{2} + b x + c = 0$ has two real roots $α$ and $β$ , where $α < - 1$ and $β > 1$ , then show that $1 + c / a + ∣ b / a ∣ < 0$ .

Let $a, b, c$ be real numbers with $a \neq = 0$ and let $α, β$ be the roots of the equation $a x^{2} + b x + c = 0$ . Express the roots of $a^{3} x^{2} + ab c x + c^{3} = 0$ in terms of $α, β$ .

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

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

In a triangle $P QR, ∠ R = π /2$ . If $tan (P /2)$ and $tan (Q /2)$ are the roots of $a x^{2} + b x + c = 0, a \neq = 0$ then

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