Solve for x in the exponential equation:
22x−9⋅2x+8=0
Get the complete, step-by-step math solution for: "Solve for x in the exponential equation: 2^{2x} - 9 · 2^x + 8 = 0". Powered by SolveForX AI math tutor.
Step-by-Step Solution
Step 1: Rewrite the equation
We can rewrite the term 22x as (2x)2 using the exponent rule (am)n=amn. This helps us see a pattern in the equation.
Step 2: Substitute to form a quadratic equation
To simplify the equation, we can substitute y for 2x. This transforms the exponential equation into a standard quadratic equation in terms of y.
Step 3: Factor the quadratic equation
We need to find two numbers that multiply to 8 and add up to -9. These numbers are -1 and -8. So, we can factor the quadratic equation as (y−1)(y−8)=0.
Step 4: Solve for y
For the product of two factors to be zero, at least one of the factors must be zero. This gives us two possible values for y: y=1 or y=8.
Step 5: Substitute back and solve for x
Now we substitute back 2x for y. For 2x=1, we know that any non-zero number raised to the power of 0 is 1, so x=0. For 2x=8, since 8=23, we have x=3.