A square mirror has sides measuring 2 ft less than the sides of a square painting. If the differ

step 1 from the question if we take square measure side as x then the painting side will be x plus 2 so

A square mirror has sides measuring 2 ft less than the sides...

A square mirror has sides measuring $2 \mathrm{ft}$ less than the sides of a square painting. If the difference between their areas is $32 \mathrm{ft}^{2},$ find the lengths of the sides of the mirror and the painting.

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Key Concepts

Variable Representation in Problem Solving

Using algebraic variables to represent unknown quantities is fundamental when translating real-world problems into mathematical expressions. In this context, a variable is assigned to the side length of one geometric figure, and relationships between figures (such as one side length being a fixed amount less than another) are modeled algebraically.

Area Calculation of a Square

Understanding that the area of a square is calculated by squaring its side length is crucial. This concept allows one to form expressions for the areas of geometric figures based on their side lengths, which is essential when comparing differences between areas.

Binomial Expansion

Expanding a binomial, such as (x - 2)², is an important algebraic technique. It converts products of binomial expressions into a sum involving squared terms and linear terms, making it easier to set up and simplify equations derived from geometric scenarios.

Formulating and Solving Equations

Translating relationships and differences expressed in words into algebraic equations is a critical problem-solving strategy. Once the equation is established, algebraic manipulation and solving techniques, including combining like terms and isolating variables, are applied to find the unknown quantities.

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