Question

7. If a + B = 3, a3+b3= 7, then a and B are the roots of
(A) 3x2 + 9x + 7 = 0 (B) 9x2 – 27x + 20 = 0 (C) 2x2 - 6x + 15 = 0 (D) None of these

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Answer 1

Answer:

$\boxed{\sf \: (B) \: \: {9x}^{2} - 27x + 20 = 0 \: }$

Step-by-step explanation:

Given that,

$\sf \: a + B = 3$

and

$\sf \: {a}^{3} + {B}^{3} = 7$

$\sf \: {(a + B)}^{3} - 3aB(a + B) = 7$

On substituting the value of a + B = 3, we get

$\sf \: {(3)}^{3} - 3aB(3) = 7$

$\sf \: 27 - 9aB = 7$

$\sf \: - 9aB = 7 - 27$

$\sf \: - 9aB = - 20$

$\implies\sf \: aB = \dfrac{20}{9}$

Now, quadratic equation having a and B as roots is given by

$\sf \: {x}^{2} - (a + B)x + aB = 0$

On substituting the values, we get

$\sf \: {x}^{2} - (3)x + \dfrac{20}{9} = 0$

$\sf \: \dfrac{ {9x}^{2} - 27x + 20}{9} = 0$

$\implies\sf \: {9x}^{2} - 27x + 20 = 0$

$\rule{190pt}{2pt}$

Additional Information :-

Nature of roots:

Let us consider a quadratic equation ax² + bx + c = 0, then nature of roots of quadratic equation depends upon Discriminant (D) of the quadratic equation.

If Discriminant, D > 0, then roots of the equation are real and unequal.

If Discriminant, D = 0, then roots of the equation are real and equal.

If Discriminant, D < 0, then roots of the equation are unreal or complex or imaginary.

Where,

Discriminant, D = b² - 4ac

Answer 2

EXPLANATION.

⇒ a + b = 3. - - - - - (1).

⇒ a³ + b³ = 7. - - - - - (2).

As we know that,

Formula of :

⇒ (x³ + y³) = (x + y)(x² - xy + y²).

⇒ (x² + y²) = (x + y)² - 2xy.

Using this formula in this question, we get.

⇒ (a³ + b³) = (a + b)(a² - ab + b²).

⇒ (a³ + b³) = (a + b)[(a + b)² - 2ab - ab].

⇒ (a³ + b³) = (a + b)[(a + b)² - 3ab].

Put the values in the equation, we get.

⇒ (7) = (3)[(3)² - 3ab].

⇒ 7 = 3(9 - 3ab).

⇒ 7 = 27 - 9ab.

⇒ 7 - 27 = - 9ab.

⇒ - 20 = - 9ab.

⇒ 20 = 9ab.

⇒ ab = 20/9.

Formula of quadratic polynomial.

⇒ x² - (a + b)x + ab.

Put the values in the formula, we get.

⇒ x² - (3)x + (20/9) = 0.

⇒ 9x² - 27x + 20 = 0.

∴ The roots of equation are : 9x² - 27x + 20 = 0.

Option [B] is correct answer.