%5E2%2B2*4%5Ex%3D9-4%5Ex-5.jpeg "(4^x-5)^2+2*4^x=9 4^x-5")
Solving the Equation: (4^x - 5)^2 + 2 * 4^x = 9 * (4^x - 5)
This equation might look intimidating at first glance, but with a little algebraic manipulation, we can solve for x. Here's how:
1. Simplify and Rearrange
- Expand the square: (4^x - 5)^2 = 4^(2x) - 10 * 4^x + 25
- Substitute and rearrange: 4^(2x) - 10 * 4^x + 25 + 2 * 4^x = 9 * 4^x - 45
- Combine like terms: 4^(2x) - 17 * 4^x + 70 = 0
2. Introduce a Substitution
Let's make the equation easier to work with by substituting a new variable:
- Let y = 4^x
- Substitute: y^2 - 17y + 70 = 0
3. Solve the Quadratic Equation
Now we have a simple quadratic equation. We can solve for y using the quadratic formula:
- Quadratic Formula: y = (-b ± √(b^2 - 4ac)) / 2a
- Where: a = 1, b = -17, c = 70
- Solving: y = (17 ± √((-17)^2 - 4 * 1 * 70)) / 2 * 1
- Simplify: y = (17 ± √89) / 2
This gives us two possible solutions for y:
- y1 = (17 + √89) / 2
- y2 = (17 - √89) / 2
4. Substitute Back and Solve for x
Now we need to substitute back y = 4^x and solve for x:
- For y1: 4^x = (17 + √89) / 2
- For y2: 4^x = (17 - √89) / 2
To solve for x, we'll use logarithms:
- For y1: x = log₄((17 + √89) / 2)
- For y2: x = log₄((17 - √89) / 2)
These are the two solutions for the original equation.
Important Note: Make sure to check your solutions by plugging them back into the original equation to ensure they are valid.