# CONSTRUCTING A CUBIC EQUATION WITH GIVEN ROOTS

Constructing a Cubic Equation with Given Roots :

In this section, you will learn how to construct a cubic equation with the roots given.

## Constructing a Cubic Equation with Given Roots - Practice Questions

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

x- 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 After having gone through the stuff given above, we hope that the students would have understood how to construct a cubic equation with the roots given.

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