medium importancemedium difficulty8 min

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.

In one line: Sum shapes the middle term, product fills the end.

Practice This Concept

Understand it

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.

Example

If the sum of two numbers is 13 and the product is 40, then x² - 13x + 40 = 0.

Formula and rule

If the sum is s and the product is p, then the quadratic equation is x² - sx + p = 0 when the numbers are roots of the equation.

See it worked out

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

Worked example

The sum of two numbers is 13 and their product is 40.

1Let one number be x, so the other is 13 - x.
2Then x(13 - x) = 40.
3Rearranging gives x² - 13x + 40 = 0.
4This is the quadratic equation formed from the sum and product.

Method you can reuse

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why do sum-and-product problems usually lead to x² - sx + p = 0?

Think first, then show answer

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.

Common wrong answer

Writing x² + 13x + 40 = 0 for a sum of 13.

Why it loses marks: The middle term sign must match the standard-form relation.

Don't mix this up

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

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.

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 from a pair of numbers, then solve it to find the actual values.

1
Enough for 1 mark
This means forming a quadratic equation when the problem gives the sum and product of two numbers or related quantities.
2
Add for 2 marks
Example: If the sum of two numbers is 13 and the product is 40, then x² - 13x + 40 = 0.
3
Add for 3 marks
If one number is x and the other is expressed in terms of x, sum and product conditions turn directly into a quadratic equation.

MCQ Quiz

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