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 (Hindi)

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: $\frac{b ^{2} - 4 a c}{a ^{2}} = \frac{B ^{2} - 4 A C}{A ^{2}}$

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Conceptually Similar Problems

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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 $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

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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

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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 $α, β$ .

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α^{2} β, α β^{2}

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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

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a, c

(B)

b, c

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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)

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(D)

a - b + 4 = 0

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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

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MathematicsTrigonometric Functions of Compound AnglesJEE Main 2005Moderate

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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

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a = b + c

(B)

c = a + b

(C)

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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

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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 (Hindi)

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

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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 $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

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.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 (Hindi)

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

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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 $(a c^{n})^{1/ (n + 1)} + (a^{n} c)^{1/ (n + 1)} + b = 0$

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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