Solve for x 4*x=4^(x-2)+27

Problem

4*x=4(x−2)+27

Solution

1. Rearrange the equation to set it equal to zero by subtracting 4*x from both sides.

4(x−2)−4*x+27=0

2. Test integer values for x to find a potential solution, as this is a transcendental equation involving both linear and exponential terms.

3. Substitute x=3 into the equation to check if it satisfies the equality.

4*(3)=4(3−2)+27

4. Simplify the right side of the equation.

12=4+27

12=31

Since 12≠31 x=3 is not a solution.

5. Substitute x=4 into the equation.

4*(4)=4(4−2)+27

16=4+27

16=16+27

16=43

Since 16≠43 x=4 is not a solution.

6. Substitute x=5 into the equation.

4*(5)=4(5−2)+27

20=4+27

20=64+27

20=91

Since 20≠91 x=5 is not a solution.

7. Test negative values or values closer to the constant. Let x=−5

4*(−5)=4(−5−2)+27

−20=4(−7)+27

−20=1/16384+27

This is not a solution.

8. Analyze the function ƒ(x)=4(x−2)−4*x+27 By testing values between 6 and 7 we see the exponential term grows much faster than the linear term. Testing values between x=6 and x=8 shows no integer solutions.

9. Use numerical methods or inspection to find the root. Let's test x=7

4*(7)=4(7−2)+27

28=4+27

28=1024+27

28=1051

10. Identify the root by observing the behavior of the linear and exponential functions. For x values where 4(x−2) is very small (large negative x, the equation approximates 4*x≈27

x≈27/4

x≈6.75

11. Refine the approximation for the root near 6.75 If x=6.75

4*(6.75)=27

4(6.75−2)+27=4+27

Since 4 is positive, x must be slightly less than 6.75

Final Answer

x≈6.75

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