Solve each problem by writing an equation and solving it by...

Solve each problem by writing an equation and solving it by completing the square.
The length of a rectangle is 8 in. more than its width. Find the dimensions of the rectangle if it has an area of 153 in $^{2}$.

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Key Concepts

Quadratic Equations

A quadratic equation is a second-degree polynomial equation that typically takes the form ax² + bx + c = 0. It often arises in problems where the relationships involve squared terms, and solving it can reveal one or two potential solutions depending on the discriminant.

Completing the Square

Completing the square is a technique used to transform a quadratic equation into a perfect square trinomial, making it easier to solve. This method involves adding and subtracting terms to rewrite the equation in the form (x + d)² = e, where one can then solve for x by taking square roots.

Algebraic Modeling of Geometric Problems

Algebraic modeling involves translating geometric or real-world problems into algebraic equations. By expressing dimensions and areas in terms of variables, one creates equations that capture the relationships described in the problem, allowing for systematic solving using algebraic methods.

Area of a Rectangle

The area of a rectangle is determined by multiplying its length by its width. This fundamental geometric relationship is the basis for setting up equations when the area is known and one of the dimensions is expressed in relation to the other.

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