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Mathematics

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

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

x = 2

2 x = 2 1

x = 1

Since 2 x = y and y = 4

x = 4

x = 2 2

N = 2

X = 1 and 2

  1. b) 5 2x + 4 X 5 x+1 – 125 = 0

(5 x2 + 4 X (5 x X 5 1) – 125 = 0

Let 5 x = 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= p,        p = 5

5 x  = 5 1

X = 1

5x = -25 (Not simplified)

1)            3 2x – 9 = 0

(3 x2 – 9 = 0

Let 3 x  = a

2 – 9 = 0

2 = 9

a = ±√9

a = ± 3

a = 3 or – 3

Since 3 x  = a,       when a = 3

3 x  = 3 1

X = 1

Since 3x = 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

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