91Ó°ÊÓ

When solving rational equations, identifying a common denominator is crucial. This allows you to eliminate fractions, simplifying the equation considerably. In our exercise, we had the expression \( \frac{1}{x-1} - \frac{2}{x^2} = 0 \). Here, the expressions have different denominators: \( x-1 \) and \( x^2 \). To simplify these terms into one equation, we need a common denominator.

To find this, look for the least common multiple of the different denominators. The least common multiple of \( x-1 \) and \( x^2 \) is \( x^2(x-1) \). Once determined, you multiply each term by this common denominator. This action helps eliminate the fractions:

• Multiplying \( x^2(x-1) \) with \( \frac{1}{x-1} \) results in \( x^2 \).
• Doing the same with \( \frac{2}{x^2} \) results in \( 2(x-1) \).

By removing fractions, the task converts into simpler algebraic expressions, making it easier to solve within fewer steps.

Many algebraic problems reduce to quadratic equations, taking on the form \( ax^2 + bx + c = 0 \). Our particular equation simplifies to \( x^2 - 2x + 2 = 0 \) after eliminating fractions and simplifying.

To solve any quadratic equation, there are several methods available:

• Factoring: Works well if the equation can be expressed as a product of binomials.
• Completing the Square: Involves rearranging the equation to create a perfect square trinomial.
• Quadratic Formula: A universal method where solutions can be found using \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).

In our exercise, we used the quadratic formula because factoring isn't straightforward for \( x^2 - 2x + 2 \), allowing us to find whether solutions are real or complex.

The discriminant is a key component in the quadratic formula, given by \( b^2 - 4ac \). It helps determine the nature of the roots for a quadratic equation. In the equation \( x^2 - 2x + 2 = 0 \), the coefficients are \( a = 1 \), \( b = -2 \), and \( c = 2 \).

The discriminant tells us:

• Positive Discriminant: Two real and distinct solutions.
• Zero Discriminant: One real, repeated solution (the vertex of the parabola touches the x-axis).
• Negative Discriminant: No real solutions, only complex ones (parabola does not intersect the x-axis).

In our exercise, the discriminant \( (-2)^2 - 4(1)(2) \) evaluates to \(-4\). This negative value indicates there are no real solutions, aligning with the fact that the graph of the equation does not intersect the x-axis.