Solve for x -2x+8=(0.25)^x

Problem

−2*x+8=(0.25)x

Solution

1. Rewrite the decimal 0.25 as a fraction to simplify the base of the exponential term.

0.25=1/4

2. Express the fraction as a power of 2 to make the relationship between the terms clearer.

1/4=2(−2)

3. Substitute the new base back into the original equation.

−2*x+8=(2(−2))x

4. Apply the power of a power rule (am)n=a(m*n) to the right side of the equation.

−2*x+8=2(−2*x)

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

Let *x=2

6. Evaluate both sides of the equation using the test value x=2

Left side: −2*(2)+8=−4+8=4

Right side: *2(−2*(2))=2(−4)=1/16

Since *4≠1/16*, try another value.

7. Test the value x=3

Left side: −2*(3)+8=−6+8=2

Right side: *2(−2*(3))=2(−6)=1/64

Since *2≠1/64*, try a smaller value.

8. Test the value x=1

Left side: −2*(1)+8=6

Right side: *2(−2*(1))=2(−2)=1/4

Since *6≠1/4*, try *x=4* to see the trend.

9. Observe the behavior of the functions ƒ(x)=−2*x+8 and g(x)=0.25 At x=4 the linear part is 0 and the exponential part is positive.

Left side: −2*(4)+8=0

Right side: *0.25=0.00390625

10. Identify the intersection point by checking values between 3 and 4 If x=3.5

Left side: −2*(3.5)+8=1

Right side: *(0.25)3.5=(2(−2))3.5=2(−7)=1/128

11. Conclude that the solution is approximately x≈3.998 using numerical methods, but for basic algebraic tasks, we look for exact intersections. Testing x=3 and x=4 shows the functions cross very close to x=4 However, checking x=3 again:

−2*(3)+8=2

(0.25)3=0.015625

The only integer-like solution occurs when the linear side equals the exponential side. By inspection, if x=3.5 the left side is 1 If x=3.99 the left side is 0.02 If x=4 the left side is 0

Final Answer

x≈3.998

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