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Quadratic Polynomial vs Quadratic Equation

A quadratic polynomial is an expression of degree 2, while a quadratic equation is that expression written equal to zero.

In one line: Polynomial is the form, equation is the fight for zero.

Practice This Concept

Understand it

A polynomial may look like x² - 4x + 3, but it becomes a quadratic equation only when it is set equal to zero, such as x² - 4x + 3 = 0. This difference matters because roots are found from the equation, not from the expression alone.

Example

x² - 5x + 6 is a quadratic polynomial, but x² - 5x + 6 = 0 is a quadratic equation.

Formula and rule

Quadratic polynomial: ax² + bx + c, with a ≠ 0.

Quadratic equation: ax² + bx + c = 0, with a ≠ 0.

See it worked out

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

Worked example

Take x² - 4x + 3.

1By itself it is a polynomial.
2If we write x² - 4x + 3 = 0, then we can solve it by factorisation: (x - 1)(x - 3) = 0, so the roots are 1 and 3.

Method you can reuse

1Check whether the expression has an equals sign.
2See whether the highest power of x is 2.
3If it is written as '... = 0', treat it as a quadratic equation.
4If not, it is only a quadratic polynomial.

Sketch it in your answer

Use a simple split: expression on one side, '= 0' on the other side, to show the change from polynomial to equation.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why is x² - 3x + 2 not yet a quadratic equation by itself?

Think first, then show answer

It is only a polynomial expression at that stage. It becomes a quadratic equation only when it is written equal to zero, because equations are solved by finding values that make the expression zero.

Common wrong answer

Saying x² - 3x + 2 is already an equation.

Why it loses marks: The zero condition is missing.

Don't mix this up

Students often treat a polynomial and an equation as the same thing and forget the '= 0' part.

Exam tip

If the question asks for roots, first check that the expression is written as an equation equal to zero.

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

CBSE may ask students to identify whether a given form is a polynomial, an equation, or both after rewriting.

1
Enough for 1 mark
A quadratic polynomial is an expression of degree 2, while a quadratic equation is that expression written equal to zero.
2
Add for 2 marks
Example: x² - 5x + 6 is a quadratic polynomial, but x² - 5x + 6 = 0 is a quadratic equation.
3
Add for 3 marks
This difference matters because roots are found from the equation, not from the expression alone.

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