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The equation $125 x + 45 x = 2 (27) x$

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Detailed Solution

We are given with an equation

125 x + 45 x = 2 (27) x

.
Rewriting the given equation in the form of square and cube of integers, we have

⇒ 53 x + (9 × 5) x − 2 (3 × 9) x ⇒ 53 x + 32 × 5 x = 2 3 × 32 x ⇒ 5 x 3 + 3 x 2 × 5 x − 2 3 x 3

Now, taking

5 x = a and 3 x = b

, we get

⇒ a 3 + a b 2 − 2 b 3 = 0

Rearranging the terms, we get

⇒ a 3 − b 3 + a b 2 − b 3 −

Now, by using the formula

a 3 − b 3 = (a − b) a 2 + a b + b 2

and rewriting the equation, we have

⇒ (a − b) a 2 + a b + b 2 + (a − b) b 2 − 0

Rewriting the equation again, we have

⇒ (a − b) a 2 + a b + b 2 + b 2 = 0

Adding the like terms, we have

⇒ (a − b) a 2 + a b + 2 b 2 = 0

Rewriting the equation, we have

⇒ a − b − 0 ⇒ a = b

Substituting

5 x = a

and

3 x = b

again, we have

⇒ 5 x = 3 x

Rewriting the equation, we have

⇒ 5 x 3 x = 1

By using the exponential formula,

a m b m = a b m

, we have

⇒ 53 x = 1

We know that the exponent raised to the power zero is one. So, we have

⇒ 53 x = 530

We can compare the powers if the exponents are equal. On comparing the powers, we get

⇒ x = 0

Therefore, the equation

125 x + 45 x = 2 (27) x

has one solution.

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