Exponential Equation of Quadratic Form

Some exponential equation can be reduced to quadratic form as can be seen below.

Example :  Solve the following equations.

1. a) 2 ^2x – 6 (2 ^x) + 8 = 0
2. b) 5 ^2x + 4 X 5 ^x+1 – 125 = 0
3. c) 3 ^2x – 9 = 0

Solution

1. a) 2 ^2x – 6 (2 ^x) + 8 = 0

(2 ^x)² – 6 (2 ^x) + 8 = 0

Let 2 ^x = y.

Then y² – 6y + 8 = 0

Then factorize

Y ² – 4 y – 2y + 8 = 0

Y (y – 4) -2 (y -4) = 0

(y -2) (y – 4) = 0

Y – 2 = 0 or y – 4 = 0

Y = 2 or y= 4

Y = 2, 4

Since 2 ^x = y, and y = 2

2 ^x = 2

2 ^x = 2 ¹

x = 1

Since 2 ^x = y and y = 4

2 ^x = 4

2 ^x = 2 ²

N = 2

X = 1 and 2

1. b) 5 ^2x + 4 X 5 ^x+1 – 125 = 0

(5 ^x) ² + 4 X (5 ^x X 5 ¹) – 125 = 0

Let 5 ^x = p

P ² + 4 X (p X 5) – 125 = 0

P² + 4 (5p) – 125 = 0

P² + 20p – 125 = 0

Then Factorise p² + 25p – 5p – 125 = 0

P (p + 25) -5 (p + 25) = 0

(p – 5) (p + 25) = 0

P – 5 = 0 p + 25 = 0

P = 5 or p = – 25

Since 5^x = p,        p = 5

5 ^x  = 5 ¹

X = 1

5x = -25 (Not simplified)

1)            3 ^2x – 9 = 0

(3 ^x) ² – 9 = 0

Let 3 ^x  = a

a ² – 9 = 0

a ² = 9

a = ±√9

a = ± 3

a = 3 or – 3

Since 3 ^x  = a,       when a = 3

3 ^x  = 3 ¹

X = 1

Since 3^x = a,        when a = -3

3 ^x  = – 3 (Not a solution)

EVALUATION ( USE THE DISCUSSION BOX AT THE BOTTOM TO SUBMIT YOUR ANSWER FOR DISCUSSION AND APPRAISAL)

Solve the following exponential equations.

1. a) 2 ^{2x+ 1} – 5 (2 ^x) + 2 = 0
2. b) 3 ^2x – 4 (3 ^x+1) + 27 = 0