Applications of Factoring x² + bx + c

When a real quantity like area is written as a trinomial x² + bx + c, factoring it into (x + p)(x + q) doesn't just simplify the expression — it reveals the two dimensions that multiply to give that area.

A rectangular room has area x² + 7x + 10 square feet. To find its length and width, factor the trinomial: look for two numbers that multiply to 10 and add to 7 — that's 2 and 5. So x² + 7x + 10 = (x + 2)(x + 5), meaning the room's dimensions are x + 2 and x + 5.

If you also know the actual area, substitute it in and solve for x: for example, if the area is 40 square feet, set (x + 2)(x + 5) = 40, expand, and solve the resulting polynomial equation for x. Once you have x, substitute it back into each factor to get the actual length and width.

Without factoring, x² + bx + c is just a number that changes with x — you can't see the two separate dimensions inside it. Factoring, using the same technique as finding the unknown c and finding the unknown b, splits the single expression back into the two quantities that produced it.

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