From the wording to an equation
In any word problem you first translate the situation into math: introduce notation, relate quantities with formulas, and set up an equation (or a system).
Earlier you often ended up with a linear equation. In this lesson the simplifications lead to a quadratic equation (or one that reduces to quadratic without introducing spurious roots from division).
Three big families of school problems are motion, work (joint or in parts), and mixtures and percents. Mixture problems rarely end in a quadratic, so below the focus is motion, work, geometry, and stories with a product of unknowns (tournament, percents).
A root of the equation is not necessarily the answer to the problem: you need a sanity check
It is convenient to let x be the quantity you are asked to find
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