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Product and sum context leading to quadratic equation

This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.

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Student-friendly explanation

If one number is x and the other is expressed in terms of x, sum and product conditions turn directly into a quadratic equation. This is common in number problems and helps students build equations quickly.

How to write this in exams

  1. 1

    Start with the exact idea

    This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.

  2. 2

    Then show how to use it

    1. Let one number be x. 2. Express the other using the sum. 3. Multiply to use the product condition. 4. Expand. 5. Rearrange into standard form.

  3. 3

    Add one concrete example

    If the sum of two numbers is 13 and the product is 40, then x^2 - 13x + 40 = 0.

  4. 4

    Avoid this incomplete answer

    Writing x^2 + 13x + 40 = 0 for a sum of 13 is wrong because the middle term sign must match the standard-form relation.

Definition

This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.

Example

If the sum of two numbers is 13 and the product is 40, then x^2 - 13x + 40 = 0.

Rule to remember

If the sum is s and the product is p, then the quadratic equation is x^2 - sx + p = 0 when the numbers are roots of the equation.

Memory hook

Sum shapes the middle term, product fills the end.

Examples and method

Worked example

The sum of two numbers is 13 and their product is 40. Let one number be x, so the other is 13 - x. Then x(13 - x) = 40. Rearranging gives x^2 - 13x + 40 = 0. This is the quadratic equation formed from the sum and product.

Method to apply

1. Let one number be x. 2. Express the other using the sum. 3. Multiply to use the product condition. 4. Expand. 5. Rearrange into standard form.

Diagram support

A two-number table with sum and product columns can guide students to the correct equation.

How CBSE asks it

Students may be asked to form the equation from a pair of numbers, then solve it to find the actual values.

Avoid common mistakes

Common confusion

Students often confuse sum with the middle coefficient sign and product with the constant term sign.

Common wrong answer

Writing x^2 + 13x + 40 = 0 for a sum of 13 is wrong because the middle term sign must match the standard-form relation.

Exam tip

For two numbers x and y, remember: sum becomes the x-term and product becomes the constant term after standard form is made.

Quick check

Why do sum-and-product problems usually lead to x^2 - sx + p = 0?

Because if the numbers are x and s - x, their product expands to a quadratic. In standard form, the sum appears as the coefficient of x and the product becomes the constant term.

Answer writing and exam use

1-mark answer

This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.

2-mark answer

This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities. If the sum is s and the product is p, then the quadratic equation is x^2 - sx + p = 0 when the numbers are roots of the equation. If the sum of two numbers is 13 and the product is 40, then x^2 - 13x + 40 = 0.

3-mark answer

If one number is x and the other is expressed in terms of x, sum and product conditions turn directly into a quadratic equation. This is common in number problems and helps students build equations quickly. If the sum is s and the product is p, then the quadratic equation is x^2 - sx + p = 0 when the numbers are roots of the equation. The sum of two numbers is 13 and their product is 40. Let one number be x, so the other is 13 - x. Then x(13 - x) = 40. Rearranging gives x^2 - 13x + 40 = 0. This is the quadratic equation formed from the sum and product. Students may be asked to form the equation from a pair of numbers, then solve it to find the actual values. Writing x^2 + 13x + 40 = 0 for a sum of 13 is wrong because the middle term sign must match the standard-form relation.
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