Easy exponentials

Algebra Level 2

0.5 x 2 − 5 x = 16 { 0.5 }^{ { x }^{ 2 }-5x }=16

Let a a and b b be the roots of the above equation.

Find the value of ∣ a − b ∣ \\ \left| a-b \right|


The answer is 3.

Ponhvoan Srey by Ponhvoan Srey , 21, Singapore

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2 solutions

Ponhvoan Srey
Oct 6, 2015

Convert the base:

0.5 x 2 − 5 x = 16 2 − ( x 2 − 5 x ) = 2 4 ⟹ x 2 − 5 x + 4 = 0 \quad \quad \quad \quad \quad { 0.5 }^{ { x }^{ 2 }-5x }=16\\ \\ \quad \quad \quad \quad \quad { 2 }^{ -({ x }^{ 2 }-5x) }={ 2 }^{ 4 }\\ \\ \Longrightarrow { x }^{ 2 }-5x+4=0

⟹ x = 1 , x = 4 ∣ a − b ∣ = 3 \\ \Longrightarrow x=1,\quad x=4\\ \\ \boxed { \left| a-b \right| =3 }

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Azadali Jivani
Oct 7, 2015

0.5^(-4) = 16
so -4 = x^2 - 5x
or x^2 - 5x + 4 = 0
x = 4 or 1
4 -1 = 3



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