Finding the Value of x³ from an Algebraic Equation

We are given the equation:\[ \frac{x}{8} + \frac{8}{x} = 1 \]The goal is to find the value of \(x^3\).

Solving the Given Algebraic Equation

To solve the equation, we first eliminate the denominators. We can do this by multiplying the entire equation by the least common multiple of the denominators, which is \(8x\), assuming \(x \neq 0\). If \(x=0\), the original equation is undefined.

Multiply both sides by \(8x\):\[ 8x \left( \frac{x}{8} + \frac{8}{x} \right) = 1 \cdot (8x) \]\[ 8x \left( \frac{x}{8} \right) + 8x \left( \frac{8}{x} \right) = 8x \]\[ x^2 + 64 = 8x \]Now, rearrange the terms to form a standard quadratic equation by moving all terms to one side:

\[ x^2 - 8x + 64 = 0 \]

This is a quadratic equation in the form \(ax^2 + bx + c = 0\), where \(a=1\), \(b=-8\), and \(c=64\).

Connecting the Quadratic Equation to Finding x³

We need to find \(x^3\). Let's consider the sum of cubes algebraic identity:\[ a^3 + b^3 = (a+b)(a^2 - ab + b^2) \]

If we compare the quadratic equation \(x^2 - 8x + 64 = 0\) with the term \((a^2 - ab + b^2)\) in the identity, it looks very similar. If we let \(a=x\) and \(b=8\), the term \((a^2 - ab + b^2)\) becomes \((x^2 - x \cdot 8 + 8^2)\), which is \((x^2 - 8x + 64)\).

So, using the identity with \(a=x\) and \(b=8\), we have:\[ x^3 + 8^3 = (x+8)(x^2 - 8x + 64) \]

We know from our algebraic manipulation that \(x^2 - 8x + 64 = 0\). Substitute this into the identity:

\[ x^3 + 8^3 = (x+8) \cdot 0 \]\[ x^3 + 512 = 0 \]

Now, we can solve for \(x^3\):

\[ x^3 = -512 \]

Thus, the value of \(x^3\) is -512.

This solution shows how a seemingly simple equation leads to a quadratic form that, while having complex roots, directly relates to an algebraic identity to find the required power of x.

Summary of Steps

• Start with the given equation \(\frac{x}{8} + \frac{8}{x} = 1\).
• Multiply by \(8x\) to clear denominators, resulting in \(x^2 + 64 = 8x\).
• Rearrange into a quadratic equation \(x^2 - 8x + 64 = 0\).
• Recognize that this quadratic expression is part of the algebraic identity for the sum of cubes: \(x^3 + 8^3 = (x+8)(x^2 - 8x + 64)\).
• Substitute \(x^2 - 8x + 64 = 0\) into the identity: \(x^3 + 8^3 = (x+8)(0)\).
• Simplify to find \(x^3 + 512 = 0\).
• Solve for \(x^3\): \(x^3 = -512\).

Revision Table: Key Algebraic Concepts

Additional Information: Complex Roots and the Problem

It's worth noting that the quadratic equation \(x^2 - 8x + 64 = 0\) has a negative discriminant (\((-8)^2 - 4(1)(64) = 64 - 256 = -192\)). This means the roots for \(x\) are complex numbers.

The roots are \(x = \frac{8 \pm \sqrt{-192}}{2} = \frac{8 \pm \sqrt{64 \cdot -3}}{2} = \frac{8 \pm 8i\sqrt{3}}{2} = 4 \pm 4i\sqrt{3}\).

Even though \(x\) is a complex number, its cube, \(x^3\), turns out to be a real number, -512, as shown using the algebraic identity. This demonstrates that sometimes, working with related algebraic identities can simplify finding values of powers of variables, even if the variable itself is complex.