algebra 2 problem

Question

How is this solve with the exact solution and decimal approximation? In √x+6=2

Answer 1

I assume you mean

 

ln √(x + 6) = 2

 

 

This is a natural log equation.  The solution to a natural log is the exponent of the base number e.

 

e ≈ 2.718

 

 

e² = √(x + 6)

 

 

Next, square both sides of the equation to undo the square-root symbol.

 

e⁴ = x + 6

 

 

Subtract 6 on both sides of the equation.

 

e⁴ - 6 = x   ----->  exact solution

 

(2.718⁴ - 6 = x 

 

48.576 = x  -----> approximate solution

 

 

 

 

 

Answer 2

Hey Lewis, let's rewrite without radical (square root sign)

 

ln (x+6)^1/2 = 2  

 

(use power property)

(Bring the power to the front)

 

ln (x+6)  =   2

    2

 

ln (x+6)  =  4

 

e^{ln (x+6)}  =  e⁴

 

x+6 = e⁴

 

x = e⁴  - 6

 

 

 

OR use inverse:

 

 

 

e^{ln (x+6)1/2} = e²

 

(x+6)^1/2  =  e²

(square both sides)

 

(x+6)  =  (e²)²

 

(x+6)  =  e⁴

 

x  =  e⁴ - 6

 

Let's check it:

 

ln (e⁴ - 6  + 6)^1/2  =  2

 

ln (e⁴)^1/2  =   2

 

ln (e²)  =  2

 

[bring 2 (power) to the front] (ln e cancel each other out)

 

2  =  2

 

It works.

 

I hope you understand both ways.  Good luck to you.

 

Yohan Chu   (LV  NV)