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medium importancemedium difficulty7 min

Geometric area context leading to quadratic equation

This means forming a quadratic equation from area-based geometry information such as rectangles, squares, or borders.

In one line: Area multiplies sides, and multiplication makes x2.

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Understand it

Area problems often multiply two expressions, so the final equation naturally becomes quadratic. The student must translate the dimensions correctly and then simplify carefully.

Example

If a rectangle has length x + 7 and breadth x, and area 60, then x(x + 7) = 60.

Formula and rule

Use area = length × breadth for rectangles or side × side for squares.

After expansion, move all terms to one side.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

A rectangle has length x + 7 cm and breadth x cm.

  1. 1Its area is 60 cm2.
  2. 2So x(x + 7) = 60.
  3. 3Expanding gives x2 + 7x - 60 = 0.
  4. 4That is the quadratic equation formed from the area context.

Method you can reuse

  1. 1Draw or imagine the shape.
  2. 2Write the expressions for the sides.
  3. 3Use the area formula.
  4. 4Expand the product.
  5. 5Rearrange into standard form.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why do area questions often lead to quadratic equations?

Think first, then show answer
Because area usually involves multiplying two side lengths. When those side lengths contain x, the expansion often gives an x2 term, so the equation becomes quadratic.

Common wrong answer

Writing x(x + 7) = 60 and stopping is incomplete because the equation should be expanded and reduced.

Don't mix this up

Students may multiply correctly but forget that the area equation must be rearranged into standard form.

Exam tip

Use a quick sketch for the shape, label the sides, and then write the area relation before expanding.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Students may be asked to form the equation first, then solve for the side length and finally find the missing dimensions.
  1. 1

    Enough for 1 mark

    This means forming a quadratic equation from area-based geometry information such as rectangles, squares, or borders.

  2. 2

    Add for 2 marks

    Example: If a rectangle has length x + 7 and breadth x, and area 60, then x(x + 7) = 60.

  3. 3

    Add for 3 marks

    Area problems often multiply two expressions, so the final equation naturally becomes quadratic.

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