Solving the Algebraic Equation \(x^4 = x^2 + 1\) for \(2x^4\)
Problem Analysis
We are given an algebraic equation \(x^4 = x^2 + 1\), with the condition that \(x > 0\). The goal is to determine the value of the expression \(2x^4\). This problem involves solving an equation that can be simplified by treating it as a quadratic equation.
Step-by-Step Solution
1. Rearrange the Equation:
The given equation is \(x^4 = x^2 + 1\). We can rewrite this by moving all terms to one side:
\(x^4 - x^2 - 1 = 0\)
2. Substitute to Form a Quadratic Equation:
Notice that the equation involves \(x^4\) and \(x^2\). We can make a substitution to simplify it. Let \(y = x^2\). Since \(x > 0\), it follows that \(x^2 > 0\), so \(y\) must be positive.
Substituting \(y\) for \(x^2\), the equation becomes:
\(y^2 - y - 1 = 0\)
3. Solve the Quadratic Equation for \(y\):
We use the quadratic formula to solve for \(y\): \(y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). In this equation, \(a=1\), \(b=-1\), and \(c=-1\).
\(y = \frac{-(-1) \pm \sqrt{(-1)^2 - 4(1)(-1)}}{2(1)}\)
\(y = \frac{1 \pm \sqrt{1 + 4}}{2}\)
\(y = \frac{1 \pm \sqrt{5}}{2}\)
4. Determine the Valid Value for \(y\) (which is \(x^2\)):
We have two potential values for \(y\): \(\frac{1 + \sqrt{5}}{2}\) and \(\frac{1 - \sqrt{5}}{2}\).
- Since \(\sqrt{5}\) is approximately \(2.236\), the value \(\frac{1 - \sqrt{5}}{2}\) is approximately \(\frac{1 - 2.236}{2} = \frac{-1.236}{2} \approx -0.618\). This is negative.
- The value \(\frac{1 + \sqrt{5}}{2}\) is approximately \(\frac{1 + 2.236}{2} = \frac{3.236}{2} \approx 1.618\). This is positive.
Because we established that \(y = x^2\) must be positive (since \(x > 0\)), we choose the positive solution:
\(x^2 = \frac{1 + \sqrt{5}}{2}\)
(This value is often referred to as the golden ratio, \(\phi\).)
5. Calculate \(x^4\):
We can find \(x^4\) directly from the original equation \(x^4 = x^2 + 1\). Substitute the value we found for \(x^2\):
\(x^4 = \left(\frac{1 + \sqrt{5}}{2}\right) + 1\)
To add these, find a common denominator:
\(x^4 = \frac{1 + \sqrt{5}}{2} + \frac{2}{2}\)
\(x^4 = \frac{1 + \sqrt{5} + 2}{2}\)
\(x^4 = \frac{3 + \sqrt{5}}{2}\)
6. Calculate the Final Expression \(2x^4\):
The question asks for the value of \(2x^4\). Multiply the expression for \(x^4\) by 2:
\(2x^4 = 2 \times \left(\frac{3 + \sqrt{5}}{2}\right)\)
The 2 in the numerator cancels with the 2 in the denominator:
\(2x^4 = 3 + \sqrt{5}\)
Thus, the value of \(2x^4\) is \(3 + \sqrt{5}\).
Similar Questions
- If \(\frac{1}{a}+\frac{1}{b}=\frac{5}{6}\) and \(\frac{1}{a^2}+\frac{1}{b^2}=\frac{13}{36}\), then what is \(\frac{1}{a^3}+\frac{1}{b^3}\) equal to?
- If $\frac{\sqrt{p+x}+\sqrt{p-x}}{\sqrt{p+x}-\sqrt{p-x}} = p$, then what is $x$ equal to?
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What is the integer assigned to N?
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