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Cross-multiplication is a valuable technique used to solve proportions—equations that express one ratio equal to another, like the original problem \(3: x = x: 12\). The method simplifies the process of finding an unknown value within these ratios.

To cross-multiply, you take the numerator of one ratio and multiply it by the denominator of the other ratio and set it equal to the product of the remaining numerator and denominator. In the given exercise, cross-multiplication transformed the proportion \(3/x = x/12\) into the equation \(3 \times 12 = x \times x\), which simplifies to \(36 = x^2\). This direct approach easily turns a problem involving ratios into a more familiar form—a simple algebraic equation, in this case, a quadratic equation that can be solved for the value of \(x\).

Quadratic equations are polynomial equations of the second degree, typically in the form \(ax^2 + bx + c = 0\). Solving them involves finding the value(s) of \(x\) that make the equation true.

In the step-by-step solution for our proportion problem, after cross-multiplication, we obtained the quadratic equation \(x^2 = 36\). To solve this, we look for values of \(x\) which, when squared, give 36. Taking the square root of both sides of the equation gives us \(x = \pm6\), revealing that there are two potential solutions. It's important to consider the context of the problem when determining which solution is meaningful; if the variable represents a physical quantity that cannot be negative, we would discard the negative solution. When solving quadratic equations, other methods like factoring, completing the square, or using the quadratic formula might also be used, depending on the form of the equation.

Ratios are comparative values expressing the relationship between quantities. They can be written in several ways: as fractions, with a colon, or with the word 'to'.

In the exercise, the ratio \(3:x\) compares the number 3 to \(x\), and similarly, \(x:12\) compares \(x\) to 12. Ratios are essential in various fields such as mathematics, science, engineering, and finance as they provide a way to compare quantities in a relative sense. Understanding ratios is fundamental to working on problems involving rates, scales, and proportions. When two ratios are set equal, they form a proportion, indicating that the two comparisons are equivalent. This equivalence is the basis of many problem-solving techniques, including the cross-multiplication method used in the given exercise.