If \(x^2-\sqrt{11}x+1=0\) , then (x 3+ x -3 ) =
Solving the Equation \(x^2-\sqrt{11}x+1=0\) to Find \(x^3 + x^{-3}\)
The problem asks us to find the value of the expression \(x^3 + x^{-3}\) given the quadratic equation \(x^2-\sqrt{11}x+1=0\). The expression \(x^3 + x^{-3}\) can be rewritten as \(x^3 + \frac{1}{x^3}\).
Step 1: Simplify the Given Equation
The given equation is:
\(x^2 - \sqrt{11}x + 1 = 0\)
Since the coefficients of \(x^2\) and the constant term are both 1, we can divide the entire equation by \(x\) (assuming \(x \neq 0\)). If \(x=0\), the equation becomes \(0 - 0 + 1 = 0\), which is \(1=0\), a contradiction. Thus, \(x\) cannot be zero, and we can safely divide by \(x\).
Dividing by \(x\), we get:
\(\frac{x^2}{x} - \frac{\sqrt{11}x}{x} + \frac{1}{x} = \frac{0}{x}\)
\(x - \sqrt{11} + \frac{1}{x} = 0\)
Rearranging the terms to isolate \(x + \frac{1}{x}\):
\(x + \frac{1}{x} = \sqrt{11}\)
This relationship, \(x + \frac{1}{x} = \sqrt{11}\), is key to solving the problem.
Step 2: Find the Value of \(x^3 + \frac{1}{x^3}\)
We need to find the value of \(x^3 + \frac{1}{x^3}\). We can use algebraic identities to relate this expression to \(x + \frac{1}{x}\).
Method 1: Using the Identity \(a^3 + b^3 = (a+b)(a^2 - ab + b^2)\)
Let \(a = x\) and \(b = \frac{1}{x}\). The identity becomes:
\(x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 - x \cdot \frac{1}{x} + (\frac{1}{x})^2)\)
\(x^3 + \frac{1}{x^3} = (x + \frac{1}{x})(x^2 - 1 + \frac{1}{x^2})\)
We already know that \(x + \frac{1}{x} = \sqrt{11}\). We need to find the value of \(x^2 + \frac{1}{x^2}\).
We can find \(x^2 + \frac{1}{x^2}\) by squaring the equation \(x + \frac{1}{x} = \sqrt{11}\):
\((x + \frac{1}{x})^2 = (\sqrt{11})^2\)
\(x^2 + 2 \cdot x \cdot \frac{1}{x} + (\frac{1}{x})^2 = 11\)
\(x^2 + 2 + \frac{1}{x^2} = 11\)
\(x^2 + \frac{1}{x^2} = 11 - 2\)
\(x^2 + \frac{1}{x^2} = 9\)
Now substitute the values of \(x + \frac{1}{x}\) and \(x^2 + \frac{1}{x^2}\) into the expression for \(x^3 + \frac{1}{x^3}\):
\(x^3 + \frac{1}{x^3} = (x + \frac{1}{x})((x^2 + \frac{1}{x^2}) - 1)\)
\(x^3 + \frac{1}{x^3} = (\sqrt{11})(9 - 1)\)
\(x^3 + \frac{1}{x^3} = \sqrt{11} \cdot 8\)
\(x^3 + \frac{1}{x^3} = 8\sqrt{11}\)
Method 2: Using the Identity \((a+b)^3 = a^3 + b^3 + 3ab(a+b)\)
Let \(a = x\) and \(b = \frac{1}{x}\). The identity becomes:
\((x + \frac{1}{x})^3 = x^3 + (\frac{1}{x})^3 + 3 \cdot x \cdot \frac{1}{x} (x + \frac{1}{x})\)
\((x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})\)
We know that \(x + \frac{1}{x} = \sqrt{11}\). Substitute this value into the identity:
\((\sqrt{11})^3 = x^3 + \frac{1}{x^3} + 3(\sqrt{11})\)
Calculate \((\sqrt{11})^3\):
\((\sqrt{11})^3 = \sqrt{11} \cdot \sqrt{11} \cdot \sqrt{11} = 11\sqrt{11}\)
Substitute this back into the equation:
\(11\sqrt{11} = x^3 + \frac{1}{x^3} + 3\sqrt{11}\)
Now, solve for \(x^3 + \frac{1}{x^3}\) by subtracting \(3\sqrt{11}\) from both sides:
\(x^3 + \frac{1}{x^3} = 11\sqrt{11} - 3\sqrt{11}\)
\(x^3 + \frac{1}{x^3} = (11 - 3)\sqrt{11}\)
\(x^3 + \frac{1}{x^3} = 8\sqrt{11}\)
Both methods give the same result.
Conclusion
Given the equation \(x^2-\sqrt{11}x+1=0\), we found that \(x + \frac{1}{x} = \sqrt{11}\). Using algebraic identities, we determined that \(x^3 + x^{-3}\), which is \(x^3 + \frac{1}{x^3}\), is equal to \(8\sqrt{11}\).
Revision Table: Key Steps
| Step | Action | Result |
|---|---|---|
| 1 | Divide \(x^2-\sqrt{11}x+1=0\) by \(x\) | \(x + \frac{1}{x} = \sqrt{11}\) |
| 2 (Method 1) | Calculate \(x^2 + \frac{1}{x^2}\) from \(x + \frac{1}{x} = \sqrt{11}\) | \(x^2 + \frac{1}{x^2} = 9\) |
| 3 (Method 1) | Use identity \(a^3+b^3\) with \(a=x, b=\frac{1}{x}\) | \(x^3 + \frac{1}{x^3} = (x+\frac{1}{x})(x^2+\frac{1}{x^2}-1)\) |
| 4 (Method 1) | Substitute values into the identity | \(x^3 + \frac{1}{x^3} = (\sqrt{11})(9-1) = 8\sqrt{11}\) |
| 2 (Method 2) | Use identity \((a+b)^3\) with \(a=x, b=\frac{1}{x}\) | \((x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})\) |
| 3 (Method 2) | Substitute \(x + \frac{1}{x} = \sqrt{11}\) into identity | \((\sqrt{11})^3 = x^3 + \frac{1}{x^3} + 3\sqrt{11}\) |
| 4 (Method 2) | Solve for \(x^3 + \frac{1}{x^3}\) | \(11\sqrt{11} = x^3 + \frac{1}{x^3} + 3\sqrt{11} \implies x^3 + \frac{1}{x^3} = 8\sqrt{11}\) |
Additional Information on Solving Algebraic Expressions
Problems involving powers of \(x\) and \(\frac{1}{x}\) are common in algebra. They often start with a quadratic equation that can be manipulated to find the value of \(x + \frac{1}{x}\) or \(x - \frac{1}{x}\). Once this base value is known, higher powers like \(x^2 + \frac{1}{x^2}\), \(x^3 + \frac{1}{x^3}\), \(x^4 + \frac{1}{x^4}\), etc., can be found using standard algebraic identities.
- To find \(x^2 + \frac{1}{x^2}\) from \(x + \frac{1}{x}\), square the expression \((x + \frac{1}{x})\). \((x + \frac{1}{x})^2 = x^2 + 2 + \frac{1}{x^2}\).
- To find \(x^2 + \frac{1}{x^2}\) from \(x - \frac{1}{x}\), square the expression \((x - \frac{1}{x})\). \((x - \frac{1}{x})^2 = x^2 - 2 + \frac{1}{x^2}\).
- To find \(x^3 + \frac{1}{x^3}\) from \(x + \frac{1}{x}\), use the identity \((x + \frac{1}{x})^3 = x^3 + \frac{1}{x^3} + 3(x + \frac{1}{x})\).
- To find \(x^3 - \frac{1}{x^3}\) from \(x - \frac{1}{x}\), use the identity \((x - \frac{1}{x})^3 = x^3 - \frac{1}{x^3} - 3(x - \frac{1}{x})\).
- To find \(x + \frac{1}{x}\) from \(x - \frac{1}{x}\) (or vice versa), use the relationship \((x + \frac{1}{x})^2 = (x - \frac{1}{x})^2 + 4\).
These identities and relationships are fundamental tools for solving this type of algebraic problem efficiently.
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