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Cross multiplication is a powerful technique used to solve equations involving fractions. It allows us to eliminate the fractions and solve the equation more easily. In an equation like \( \frac{a}{b} = \frac{c}{d} \), cross multiplication involves multiplying the numerator of one fraction by the denominator of the other fraction and setting the resulting products equal. This transforms the equation into \( a \cdot d = b \cdot c \). In the original step-by-step solution, cross multiplication is used to solve: \( \frac{x(x+1) + 20}{4(x-1)(x+1)} = \frac{1}{4} \). Here, we multiply the numerators and denominators across the equal sign: 4 multiplied by \((x(x+1) + 20)\) equals \((x-1)(x+1)\) multiplied by 1. With this step, the fractions disappear, leaving behind a simpler equation: \( 4(x^2 - 1) = x^2 + x + 20 \). This eliminates fractions and so simplifies the process of solving quadratic equations substantially. The ease and efficiency of cross multiplication make it a staple strategy when dealing with fractional equations.

Using a common denominator is crucial when solving equations with multiple fractions. The common denominator brings all terms to a level playing field, allowing us to combine and simplify them easily. In this problem, the original fractions were \( \frac{x}{4x-4} \) and \( \frac{5}{x^2-1} \). The denominators are factored as \( 4(x-1) \) and \((x-1)(x+1)\). The least common denominator (LCD) combines all unique factors from each denominator, which results in \( 4(x-1)(x+1) \). Once the common denominator is identified, each fraction is rewritten with this denominator, transforming the original equation into one cohesive expression:

• \( \frac{x(x+1)}{4(x-1)(x+1)} \) represents the first term.
• \( \frac{20}{4(x-1)(x+1)} \) adjusts the second term.

The goal is to express the right side of the equation \( \frac{1}{4} \) with the common denominator, confirming that all parts of the equation share the same denominator. This unified denominator enables us to then focus exclusively on the numerators, smoothing the way for solving the equation through subsequent steps.

When an equation meets the pattern of a quadratic, namely in the form \( ax^2 + bx + c = 0 \), the quadratic formula becomes our go-to tool for finding its roots. This powerhouse formula, \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), works efficiently for any quadratic equation, providing exact solutions even when the equation cannot be easily factored.In the given solution, the equation evolved to \( 3x^2 - x - 24 = 0 \) after combining like terms and setting the equation to zero. From here, applying the quadratic formula involves:

• Identifying the coefficients: \( a = 3 \), \( b = -1 \), and \( c = -24 \).
• Calculating the discriminant: \( b^2 - 4ac = 289 \), an essential value as it determines the nature of the roots.

With a positive discriminant of 289, we proceed with solving:

• \( x = \frac{1 \pm 17}{6} \) determining that the solutions are \( x = 3 \) and \( x = -\frac{8}{3} \).

The quadratic formula not only provides solutions accurately but also offers insight into the equation's characteristics through its discriminant.