Solve each problem. See Examples 7 and 8 . A square mirror...

Solve each problem. See Examples 7 and $8 .$
A square mirror has sides measuring 2 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

Translating Word Problems into Algebraic Equations

This concept involves converting the information given in a descriptive problem into a mathematical equation. It includes identifying the unknowns and expressing relationships between different quantities, which is crucial in setting up the correct equation to solve a problem.

Area of a Square

Understanding that the area of a square is found by squaring the length of its side is fundamental. This concept is used to express the areas of both the painting and the mirror in algebraic terms, forming the basis for setting up the problem's equation.

Algebraic Expansion and Simplification

This involves expanding expressions such as (x - 2)^2 into x^2 - 4x + 4 and then simplifying equations by combining like terms. Such manipulation is vital for transforming complex expressions into simpler forms that are easier to solve.

Solving Quadratic Equations

Once the equation is set up and simplified, it may take the form of a quadratic equation. Being able to solve quadratic equations—whether by factoring, completing the square, or using the quadratic formula—is a key skill in finding the unknown values in this type of problem.

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