Question

sum equal 6 and product = - 924 find the two numbers which when added get the sun as 6and multiplied get the product as - 924 ​

Answer 1

Answer:

The two numbers are $3+\sqrt{933}$ and $(3-\sqrt{933}$

Step-by-step explanation:

Given :

To find two numbers, whose sum is 6 & product is -924.

Solution :

Let the numbers be x & y,.

Then,

x + y = 6  ..(i)

&

xy = -924 ..(ii)

We know that,

(a + b)² = a² + 2ab + b²

&

(a - b)² = a² - 2ab + b²

By using these identities,..

⇒ x + y = 6

⇒ (x + y)² = (6)²

⇒ x² + 2xy + y² = 36

⇒ (x² + y²) + 2(xy) = 36

⇒ (x² + y²) + 2(-924) = 36

⇒ (x² + y²) - 1848 = 36

⇒ (x² + y²) = 36 + 1848

⇒ x² + y² = 1884 ..(iii)

_

⇒ (x - y)² = x² - 2xy + y²

⇒ (x - y)² = (x² + y²) - 2(xy)

⇒ (x - y)² = 1884 -2(-924)

⇒ (x - y)² = 1884 + 1848

⇒ (x - y)² = 3732

⇒ x - y = √3732 ..(iv)

By adding (i) & (iv),

We get,

⇒ (i) + (iv)

⇒ (x + y) + (x - y) = 6 + √3732

⇒ 2x = 6 + √3732 =  2 × (3 + √933)

⇒ x = $\frac{2(3 +\sqrt{933}}{2} = 3 + \sqrt{933}$

By subtracting (iv) from (i),

We get,

⇒ (i) - (iv)

⇒ (x + y) - (x - y) = 6 - √3732

⇒ 2y = 6 - √3732 =  2 × (3 - √933)

⇒ y = $\frac{2(3 -\sqrt{933}}{2} = 3 - \sqrt{933}$

___

(or)

Alternate Solution :

x + y = 6 ..(i)

&

xy = -924 ..(ii)

From (i),

⇒ x + y = 6

⇒ y = 6 - x ... (iii)

From (ii) & (iii)

⇒ xy = -924

⇒ x(6 - x) = -924

⇒ 6x - x² + 924 = 0

⇒ x² - 6x - 924 = 0

By using Quadratic formula,

$x = \frac{-b±\sqrt{b^2 - 4ac} }{2a}$

where , a = 1 , b = -6 , c = -924

As, the equation is of the form, ax² + bx + c = 0

⇒ $x = \frac{-(-6)±\sqrt{(-6)^2 - 4(1)(-924)} }{2(1)}$

⇒ $x = \frac{6±\sqrt{36 + 3696} }{2}$

⇒ $x = \frac{6±\sqrt{3732} }{2}$

⇒ $x = \frac{2(3±\sqrt{933})}{2}$  (As √3732 = √4 × √933 = 2√933)

⇒  $x = 3+\sqrt{933}$ (or) $x = 3-\sqrt{933}$

⇒   $x =3+\sqrt{933}$ (or) $y = 3-\sqrt{933}$