Solving Exponential Equation \((2+\sqrt{3})^x + (2-\sqrt{3})^x = 2\)
This problem asks us to find the value of the expression \((2+\sqrt{3})^x - (2-\sqrt{3})^x\) given the equation \((2+\sqrt{3})^x + (2-\sqrt{3})^x = 2\). We can solve this using algebraic manipulation.
Defining Variables for Simplicity
To make the expressions easier to work with, let's define:
- $a = (2+\sqrt{3})^x
- $b = (2-\sqrt{3})^x
Using these definitions, the given equation can be rewritten as:
\(a + b = 2\)
And the expression we need to find is:
\(a - b\)
Analyzing the Terms \((2+\sqrt{3})^x\) and \((2-\sqrt{3})^x\)
Let's examine the base terms, \(2+\sqrt{3}\) and \(2-\sqrt{3}\). Notice that they are conjugates of each other. Let's find their product:
\((2+\sqrt{3})(2-\sqrt{3}) = 2^2 - (\sqrt{3})^2\) \(= 4 - 3\) \(= 1\)
Since the product of the bases is 1, this means:
- \(b = (2-\sqrt{3})^x = \left( \frac{1}{2+\sqrt{3}} \right)^x = \frac{1}{(2+\sqrt{3})^x} = \frac{1}{a}\)
- Alternatively, \(a = (2+\sqrt{3})^x = \left( \frac{1}{2-\sqrt{3}} \right)^x = \frac{1}{(2-\sqrt{3})^x} = \frac{1}{b}\)
- Also, the product \(ab = [(2+\sqrt{3})(2-\sqrt{3})]^x = [1]^x = 1\).
This relationship \(b = 1/a\) (or \(ab=1\)) is key to solving the equation.
Solving for the Variable Values
Substitute \(b = \frac{1}{a}\) into the given equation \(a + b = 2\):
\(a + \frac{1}{a} = 2\)
To solve for \(a\), we can clear the fraction by multiplying both sides by \(a\) (assuming \(a \neq 0\), which is true since \((2+\sqrt{3})^x\) is always positive):
\(a \cdot \left( a + \frac{1}{a} \right) = 2 \cdot a\)
\(a^2 + 1 = 2a\)
Now, rearrange this into a standard quadratic equation form:
\(a^2 - 2a + 1 = 0\)
This is a perfect square trinomial:
\((a-1)^2 = 0\)
Taking the square root of both sides gives:
\(a - 1 = 0\)
\(a = 1\)
Now that we have found \(a=1\), we can find \(b\). Using \(b = \frac{1}{a}\):
\(b = \frac{1}{1} = 1\)
So, we have \(a = (2+\sqrt{3})^x = 1\) and \(b = (2-\sqrt{3})^x = 1\).
Calculating the Target Expression \((2+\sqrt{3})^x-(2-\sqrt{3})^x\)
We need to find the value of \(a - b\). Since we found that \(a = 1\) and \(b = 1\), the calculation is straightforward:
\(a - b = 1 - 1 = 0\)
Therefore, \((2+\sqrt{3})^x - (2-\sqrt{3})^x = 0\).
Alternative Method Using \((a-b)^2\)
Another way to find \(a-b\) is to use the identity \((a-b)^2 = (a+b)^2 - 4ab\).
- We are given \(a+b=2\).
- We found \(ab=1\).
Substituting these values:
\((a-b)^2 = (2)^2 - 4(1)\)
\((a-b)^2 = 4 - 4\)
\((a-b)^2 = 0\)
Taking the square root of both sides:
\(a-b = 0\)
Both methods confirm that the value of the expression is 0.
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