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Representing real-life situations as quadratic equations

This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation.

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Student-friendly explanation

Many CBSE questions describe lengths, areas, products, or consecutive numbers in words. The student must translate the words into an equation, rearrange it into standard form, and then solve it.

How to write this in exams

  1. 1

    Start with the exact idea

    This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation.

  2. 2

    Then show how to use it

    1. Read the relation carefully. 2. Let the unknown quantity be x. 3. Write the formula from the context. 4. Expand the expression. 5. Rearrange it into ax^2 + bx + c = 0. 6. Solve using the suitable method.

  3. 3

    Add one concrete example

    If breadth is x and length is x + 5, then area 84 gives x(x + 5) = 84.

  4. 4

    Avoid this incomplete answer

    Writing x^2 + 5x = 84 and stopping there is incomplete because the equation is not yet in standard form.

Definition

This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation.

Example

If breadth is x and length is x + 5, then area 84 gives x(x + 5) = 84.

Rule to remember

Use area = length × breadth, product relations, or consecutive-number relations like x(x + 1) or x(x + 2) when the question describes real situations.

Memory hook

Words first, equation second, answer last.

Examples and method

Worked example

A rectangle has breadth x cm and length x + 5 cm. Its area is 84 cm^2. So x(x + 5) = 84. Expanding gives x^2 + 5x - 84 = 0, which is the quadratic equation to solve.

Method to apply

1. Read the relation carefully. 2. Let the unknown quantity be x. 3. Write the formula from the context. 4. Expand the expression. 5. Rearrange it into ax^2 + bx + c = 0. 6. Solve using the suitable method.

Diagram support

A quick sketch of the shape or number relation helps students see which values are multiplied or compared.

How CBSE asks it

The exam may present a practical situation and ask for the equation first, then the value of x and the final dimensions.

Avoid common mistakes

Common confusion

Students often write the relation correctly but forget to bring all terms to one side before solving.

Common wrong answer

Writing x^2 + 5x = 84 and stopping there is incomplete because the equation is not yet in standard form.

Exam tip

Choose one variable clearly, build the relation from the question, and expand carefully before solving.

Quick check

Why does a rectangle problem often lead to a quadratic equation?

Because area is found by multiplying two expressions, such as x and x + 5. After expansion, the relation usually becomes a degree-2 equation, which is quadratic.

Answer writing and exam use

1-mark answer

This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation.

2-mark answer

This means turning a word problem or geometry problem into a quadratic equation by choosing a variable and forming the correct relation. Use area = length × breadth, product relations, or consecutive-number relations like x(x + 1) or x(x + 2) when the question describes real situations. If breadth is x and length is x + 5, then area 84 gives x(x + 5) = 84.

3-mark answer

Many CBSE questions describe lengths, areas, products, or consecutive numbers in words. The student must translate the words into an equation, rearrange it into standard form, and then solve it. Use area = length × breadth, product relations, or consecutive-number relations like x(x + 1) or x(x + 2) when the question describes real situations. A rectangle has breadth x cm and length x + 5 cm. Its area is 84 cm^2. So x(x + 5) = 84. Expanding gives x^2 + 5x - 84 = 0, which is the quadratic equation to solve. The exam may present a practical situation and ask for the equation first, then the value of x and the final dimensions. Writing x^2 + 5x = 84 and stopping there is incomplete because the equation is not yet in standard form.
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