Solving the Equation 4^x - 3(2^x + 3) + 128 = 0
In this article, we will solve the equation 4^x - 3(2^x + 3) + 128 = 0. This equation involves exponential functions and requires some manipulation to solve.
Step 1: Simplify the Equation
First, let's simplify the equation by evaluating the expression inside the parentheses:
4^x - 3(2^x + 3) + 128 = 0 4^x - 3(2^x) - 9 + 128 = 0 4^x - 3(2^x) - 9 + 128 = 0
Step 2: Rewrite the Equation
Now, let's rewrite the equation by combining like terms:
4^x - 3(2^x) = -119
Step 3: Express 4^x in Terms of 2^x
Since 4 = 2^2, we can express 4^x as:
4^x = (2^2)^x = 2^(2x)
So, the equation becomes:
2^(2x) - 3(2^x) = -119
Step 4: Substitute y = 2^x
Let's substitute y = 2^x to simplify the equation:
y^2 - 3y = -119
Step 5: Solve the Quadratic Equation
Now, we have a quadratic equation in y. Let's solve it using the quadratic formula:
y = (-b ± √(b^2 - 4ac)) / 2a
In this case, a = 1, b = -3, and c = -119. Plugging these values into the formula, we get:
y = (3 ± √((-3)^2 - 4(1)(-119))) / 2(1) y = (3 ± √(9 + 476)) / 2 y = (3 ± √485) / 2
Simplifying the expression, we get two possible values for y:
y = (3 ± 22) / 2
y = 25/2 or y = -19/2
Step 6: Find the Value of x
Now that we have the values of y, we can find the value of x:
2^x = y x = log2(y)
For y = 25/2, we get:
x = log2(25/2) ≈ 4.29
For y = -19/2, we get:
x = log2(-19/2) is undefined (since the logarithm of a negative number is undefined in real numbers)
Therefore, the only solution to the equation is x ≈ 4.29.
