CONSTRUCTING A CUBIC EQUATION WITH GIVEN ROOTS

Question 1 :

If the sides of a cubic box are increased by 1, 2, 3 units respectively to form a cuboid, then the volume is increased by 52 cubic units. Find the volume of the cuboid.

Solution :

Let x be the side length of the cube 

Volume of cube is

=  x³

Length of cuboid  =  (x + 1)

Breadth of cuboid  =  (x + 2)

Height of cuboid  =  (x + 3)

Volume of cuboid  =  (x + 1) (x + 2) (x + 3)

(x + 1) (x + 2) (x + 3)  =  52

(x² + 3x + 2) (x + 3)  =  52

x³ + 3x² + 3x² + 9x + 2x + 6  =  x³ + 52

x³ - x³ + 6x² + 11x + 6 - 52  =  0

6x² + 11x - 46  =  0

6x² - 12x + 23x  - 46  =  0

6x (x - 2) + 23(x - 2)  =  0

6x + 23  =  0, x - 2  =  0

x = 2

Volume of cube is

=  x³

=  2³

=  8 cubic units.

Question 2 :

Construct a cubic equation with roots.

(i) 1, 2 and 3 (ii) 1,1, and −2 (iii) 2, 1/2 and 1.

Solution :

(i) 1, 2 and 3 

x = 1, x = 2 and x = 3

(x - 1) (x - 2) (x - 3)  =  0

(x² - 2x - x + 2)(x - 3)  =  0

(x² - 3x + 2)(x - 3)  =  0

x³ - 3x² - 3x² + 9x + 2x - 6  =  0

x³ - 6x² + 11x - 6  =  0

(ii) 1, 1 and −2

x = 1, x = 1 and x = -2

(x - 1) (x - 1) (x + 2)  =  0

(x² - x - x + 1)(x + 2)  =  0

(x² - 2x + 1)(x + 2)  =  0

x³ + 2x² - 2x² - 4x + 1x + 2  =  0

x³  - 3x + 2  =  0

(iii) 1, 1/2 and 1

x = 1, x = 1/2 and x = 1

(x - 1) (x - 1/2) (x - 1)  =  0

(x - 1)²(x - 1/2)  =  0

(x² - 2x + 1) (2x - 1)/2)  =  0

2x³ - x² - 4x² + 2x + 2x - 1  =  0

2x³ - 5x² + 4x - 1  =  0

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