Represent the following situations in the form of quadratic...

Represent the following situations in the form of quadratic equations: (i) The area of a rectangular plot is $528 \mathrm{~m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot. (ii) The product of two consecutive positive integers is 306 . We need to find the integers. (iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360 . We would like to find Rohan's present age. (iv) A train travels a distance of $480 \mathrm{~km}$ at a uniform speed. If the speed had been $8 \mathrm{~km} / \mathrm{h}$ less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Represent the following situations in the form of quadratic equations:
(i) The area of a rectangular plot is $528 \mathrm{~m}^{2}$. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306 . We need to find the integers.
(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360 . We would like to find Rohan's present age.
(iv) A train travels a distance of $480 \mathrm{~km}$ at a uniform speed. If the speed had been $8 \mathrm{~km} / \mathrm{h}$ less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.

Key Concepts

Quadratic Equations

A quadratic equation is a second-degree equation in one variable, generally expressed in the standard form ax² + bx + c = 0, where a, b, and c are constants with a ? 0. Quadratic equations are pivotal in algebra as they model various problems involving area, motion, and number relationships, and they often yield two solutions corresponding to different scenarios in applied contexts.

Translating Word Problems into Equations

This concept involves interpreting language-based problem statements to form mathematical expressions or equations. In many real-world problems, conditions provided in words need to be converted into algebraic equations by assigning variables, setting up relationships, and forming quadratic expressions when the conditions involve products of unknowns or squared terms.

Modeling Geometric Relationships with Algebra

Many geometric problems, such as those involving the area of a rectangle, naturally lead to quadratic equations when one dimension is expressed in terms of another. This key concept illustrates how geometric properties, like area, can be formulated using algebraic relationships, facilitating the transition from spatial concepts to solvable equations.

Formulating Equations from Consecutive Number Relationships

Problems involving consecutive integers or numbers often give rise to quadratic equations because the structure of consecutive numbers (differing by a constant) leads to algebraic expressions that, when multiplied or added, form second-degree polynomials. Recognizing the pattern allows for setting up and solving the relevant quadratic equation.

Uniform Motion and Time-Distance-Speed Relationships

In uniform motion problems, relationships between speed, distance, and time are fundamental. When variations in speed yield differences in travel time, these relationships can be expressed in equations that involve quadratic forms. Understanding how to set up and manipulate these equations is essential for accurately modeling and solving motion-related problems.

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