Determine Nature of Roots for Quadratic Equation

First, convert the given equation into the standard quadratic form, $ax^2 + bx + c = 0$.

Given equation: $\text{x} - \frac{1}{\text{x}} = 17$.

• Multiply by $x$ (assuming $x \neq 0$): $x(x) - x(\frac{1}{x}) = 17x$ $x^2 - 1 = 17x$
• Rearrange into standard form: $x^2 - 17x - 1 = 0$

Identify the coefficients $a$, $b$, and $c$ from $x^2 - 17x - 1 = 0$:

• $a = 1$
• $b = -17$
• $c = -1$

Calculate Discriminant to Find Root Nature

The nature of the roots depends on the discriminant, calculated using the formula $\Delta = b^2 - 4ac$.

• Calculate $\Delta$: $\Delta = (-17)^2 - 4(1)(-1)$ $\Delta = 289 - (-4)$ $\Delta = 289 + 4$ $\Delta = 293$

Analyze Discriminant Result

Analyze the value of the discriminant, $\Delta = 293$.

• Since $\Delta = 293 > 0$, the roots are real and distinct.
• To determine if they are rational or irrational, check if $\Delta$ is a perfect square. $17^2 = 289$ and $18^2 = 324$. Since 293 is between these squares, it is not a perfect square.
• Therefore, the distinct real roots are irrational.

Conclusion: The roots of the equation $\text{x} - \frac{1}{\text{x}} = 17$ are real and distinct (irrational).