- \( 84 x=21 x^{2} \)

To solve the equation \( 84x = 21x^2 \), start by rearranging it to one side: \( 21x^2 - 84x = 0 \). Factor out the common term \( 21x \), leading to \( 21x(x - 4) = 0 \). This gives two solutions: \( x = 0 \) or \( x = 4 \). So, we find that the solutions are \( x = 0 \) and \( x = 4 \). An interesting twist in mathematics is that equations like this can have many applications, from physics to economics. For instance, in physics, such equations might represent relationships between quantities, like force and area, which can help solve real-world problems, such as calculating pressure. However, when solving quadratic equations, a common mistake is to forget to check for all possible solutions, especially in cases where factoring may lead to extraneous solutions. Make sure to substitute your answers back into the original equation to confirm they are valid solutions!