If one of the roots of equation x2 + mx + n = 0 is the square of the o

1.	If one of the roots of equation x2 + mx + n = 0 is the square of the other, then which of the following condition will be true? 1. m3 + n2 + n = -3mn2. m3 + n2 + n = 3mn3. n3 + m2 + m = 3mn4. n3 + m2 + m = -3mn
Answer» Correct Answer - Option 2 : m³ + n² + n = 3mn Given: One of the roots of equation x² + mx + n = 0 is the square of the other. Formula Used: If ax² + bx + c = 0, then Sum of the roots = -(b/a) Product of the roots = (c/a) (a + b)³ = a³ + b³ + 3 × a × b × (a + b) Calculation: Let α and α² be the roots of equation x² + mx + n = 0 Product of the roots = (c/a) ⇒ α × α² = n ⇒ α³ = n ⇒ α = (n)^1/3 ---- (1) Sum of the roots = -(b/a) ⇒ α + α² = -m ⇒ α + α² = -m ---- (2) Put the value of eq. (1) in the eq. (2) ⇒ (n)^1/3 + (n)^2/3 = -m On cubing both the sides, ⇒ n + n² + 3 × n^1/3 × n^2/3 × (n^1/3 + n^2/3) = (-m)³ ⇒ n + n² + 3 × n × (-m) = -m³ ⇒ n + n² + m³ = 3mn ∴ The condition n + n² + m³ = 3mn will be true.

If one of the roots of equation x2 + mx + n = 0 is the square of the other, then which of the following condition will be true? 1. m3 + n2 + n = -3mn2. m3 + n2 + n = 3mn3. n3 + m2 + m = 3mn4. n3 + m2 + m = -3mn

Answer» Correct Answer - Option 2 : m³ + n² + n = 3mn

Given:

One of the roots of equation x² + mx + n = 0 is the square of the other.

Formula Used:

If ax² + bx + c = 0, then

Sum of the roots = -(b/a)

Product of the roots = (c/a)

(a + b)³ = a³ + b³ + 3 × a × b × (a + b)

Calculation:

Let α and α² be the roots of equation x² + mx + n = 0

Product of the roots = (c/a)

⇒ α × α² = n

⇒ α³ = n

⇒ α = (n)^1/3 ---- (1)

Sum of the roots = -(b/a)

⇒ α + α² = -m

⇒ α + α² = -m ---- (2)

Put the value of eq. (1) in the eq. (2)

⇒ (n)^1/3 + (n)^2/3 = -m

On cubing both the sides,

⇒ n + n² + 3 × n^1/3 × n^2/3 × (n^1/3 + n^2/3) = (-m)³

⇒ n + n² + 3 × n × (-m) = -m³

⇒ n + n² + m³ = 3mn

∴ The condition n + n² + m³ = 3mn will be true.