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In the realm of algebra, finding the roots of an equation, especially a quadratic one, is akin to solving a puzzle. The roots are solutions that make the equation equal zero. For example, in the equation \( ax^2 + bx + c = 0 \), the roots are those specific values of \( x \) that satisfy this condition. When we refer to "finding the roots," we're essentially looking for where the graph of the equation intersects the x-axis. This is significant because these intersections represent points where the outcome of the equation is nothing, or zero.

Not all equations will have real roots; some might have complex ones. However, in our original exercise, we deal with potential real roots. By substituting a potential root, like \( x = \frac{5}{4} \), directly into the equation, we check if this value satisfies the condition of making the entire equation zero. The process is simple in concept—one puts the value into the equation and simplifies to see if the result equals zero—but it requires careful calculation at each step.

Understanding whether a specific value is indeed a root involves testing it accurately through substitution and simplification. This ensures no computational errors skew the final assessment.

The substitution method is a powerful tool in solving equations, and it's especially useful when checking whether a given value is a root of an equation. The method is all about replacing variables with numbers or expressions to see if an equality holds true. It’s like replacing the scoop of the puzzle piece to see if it fits into the right spot.

In the context of our quadratic equation \(12x^2 - x - 20 = 0\), we substitute \( x = \frac{5}{4} \) into the equation. This is straightforward—every instance of \( x \) in the equation is replaced with \(\frac{5}{4}\). However, this substitution must be done with precision:

• First, substitute the value: \(12\left(\frac{5}{4}\right)^2 - \frac{5}{4} - 20\).
• Next, carefully follow the order of operations—first squaring the fraction, then multiplying, and finally simplifying each term.

The substitution method primarily allows us to test assumptions and verify solutions by transforming the equation with specific values. It acts as a necessary step to clearly see if our potential root is effective.

Simplification is at the heart of solving and verifying solutions in mathematics. It involves reducing expressions to their simplest form, which makes them easier to work with and understand. For the quadratic equation example, simplification helped us assess whether \( x = \frac{5}{4} \) was indeed a root.

Let’s break down the simplification process detailed in the solution:

• First, perform any operations within the equation such as squaring \(\left(\frac{5}{4}\right)^2\) to get \(\frac{25}{16}\).
• Next, handle multiplication: \(12 \times \frac{25}{16}\) resulting in \(\frac{75}{4}\).
• Finally, combine and reduce like terms: \(\frac{75}{4} - \frac{5}{4} - 20\).

By converting 20 into a fraction \(\frac{40}{2}\) and subtracting from \(\frac{35}{2}\), it became clear through simplification that the equation didn't equal zero. Thus, \( x = \frac{5}{4} \) was not a root.

Such simplification steps are critical because they uncover the truth about whether a potential solution solves the equation by reducing all complexity, allowing us to focus solely on solving the problem.