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Applications of polynomials: x^2 + bx + c

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Applications of Factoring x² + bx + c

When area or another real quantity is written as x squared plus bx plus c, factoring it into (x+p)(x+q) reveals the two dimensions multiplied to create it. Learn to factor a trinomial application, solve for an unknown using the actual value, and recover the real-world dimensions.

Using factoring to solve applications

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.

Using factoring to find dimensions A rectangular room has area x squared plus 7x plus 10. Factoring the area as (x+2)(x+5) reveals the room's length and width in terms of x. Area = x² + 7x + 10 Factor: x² + 7x + 10 = (x + 2)(x + 5) length = x + 5 width = x + 2
Factoring the area x²+7x+10 as (x+2)(x+5) reveals the room's length and width.

Worked example: finding room dimensions

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.

Solving for a numeric answer

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.

Why factoring helps here

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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