Paper 4: Fundamentals of Business Mathematics & Statistic

2.54 I FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS

Algebra

Example 95 : Solve x 46 x 2 −− =x 2x+ +^2

Solution :

Multiplying by the L.C.M. of the denominators, we find :

6 - x = x (x – 2) + 2 (x^2 – 4) or, 3x^2 – x – 14 = 0

or, (3x – 7) (x + 2) = 0

∴ either 3x – 7 = 0 or, x + 2 = 0

x^7

∴ = 3 or, – 2

Now x = – 2 does not satisfy the equation, x=^73 is the root of the equation.

Example 96: Solve 4x – 3.2x + 2 + 2^5 = 0.

Solution :

Here, 22x – 3.2x .2^2 + 32 = 0

or, (2x)^2 – 12.2x + 32 = 0 or, u^2 – 12u + 32 = 0 (taking u = 2x)

or, (u – 4) (u – 8) = 0

∴ either (u – 4) = 0 or, (u – 8) = 0 ∴ u = 4, 8

When u = 4, 2x = 4 = 2^2 ∴ x = 2

Again u = 8, 2x = 8 = 2^3 ∴ x = 3.

Extraneous Solutions :

In solving equations, the values of x so obtained may not necessarily be the solution of the original equation.

Care should be taken to verify the roots in each case, and also square roots (unless stated otherwise) are to

be taken as positive.

Example 97: Solve x 7x x 7x 9 3^2 + +^2 + + =

Solution :

Adding 9 to both sides, we have x 7x 9 x 7x 9 12^2 + + +^2 + + =

Now putting u x 7x 9,=^2 + + the equation reduces to

u^2 + u – 12 = 0

or, u^2 + 4u – 3u – 12 = 0 or, u (u + 4) – 3 (u + 4) = 0

or, (u – 3) (u + 4) = 0 ∴ u = 3, – 4.

Since u is not negative, we reject the value – 4 for u.

When u = 3, x +7x +9 = 3^2 or, x^2 + 7x + 9 = 9

or, x (x + 7) = 0 or, x = 0, – 7.

Example 98 : Solve (x^2 + 3x)^2 + 2 (x^2 + 3x) = 24

Solution :

Let x^2 + 3x = u, so that equ. becomes u^2 + 2u = 24 or, u^2 + 2u – 24 = 0

or, u^2 + 6u – 4u – 24 = 0 or, u (u + 6) – 4 (u + 6) = 0