Given x^2-3x-1=0, what is 1/x-x ?

Given x^2-3x-1=0, what is 1/x-x ?

By factoring x^2 – 3x – 1 = 0, the solutions for x are -1 and 1. Substituting these values into the expression 1/x – x gives:

When x = 1, 1/x – x = 1 – 1 = 0

When x = -1, 1/x – x = -1 – (-1) = 0

Therefore, with the solutions of x = -1 and x = 1, the simplified expression 1/x – x equals 0.

Using Factorization to Simplify Expressions

When presented with an expression containing variables, being able to simplify it can reveal useful information and make it easier to manipulate. This often involves solving for the variables. Let’s look at the step-by-step process for simplifying 1/x – x given the equation x^2 – 3x – 1 = 0.

First, we need to solve for x using the quadratic formula or by factoring the equation x^2 – 3x – 1 = 0.

The factors of the expression are (x – 1) and (x + 1).

Setting each factor equal to 0 gives the solutions x = 1 and x = -1.

Now we can substitute these values for x into the provided expression 1/x – x.

When x = 1: 1/x – x 1/1 – 1 1 – 1 = 0

When x = -1: 1/x – x 1/-1 – (-1) -1 – (-1) = 0

Therefore, the simplified expression 1/x – x = 0 when x is either 1 or -1.

This example demonstrates how factoring a quadratic equation can reveal the solutions for the variable x. We can then substitute these solutions into an expression to simplify and evaluate it. The factorization method provides an alternative to using the quadratic formula for solving quadratics. Factoring is faster when the expression has integer factors that are easy to identify.

Whether using factoring, the quadratic formula, or another method, finding the solutions for x allows us to gain insight into related expressions. Simplifying expressions by substituting solved variable values is an essential algebra skill. With practice, you can quickly factor equations, identify solutions, and evaluate expressions containing x. Mastering these algebraic steps develops skills for simplifying problems across mathematics, sciences, and engineering.