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Identifying a Quadratic Equation

A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0.

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Student-friendly explanation

Not every equation with x is quadratic. To identify a quadratic equation, first simplify it and check that the highest power of x is 2 and no higher power remains.

How to write this in exams

  1. 1

    Start with the exact idea

    A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0.

  2. 2

    Then show how to use it

    1. Expand brackets or remove fractions if needed. 2. Bring all terms to one side. 3. Check the highest power of x. 4. If the highest power is 2, it is quadratic.

  3. 3

    Add one concrete example

    2x(x - 3) = 8 becomes 2x^2 - 6x - 8 = 0, so it is quadratic.

  4. 4

    Avoid this incomplete answer

    Saying 3x + 5 = 0 is quadratic is wrong because the highest power is only 1.

Definition

A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0.

Example

2x(x - 3) = 8 becomes 2x^2 - 6x - 8 = 0, so it is quadratic.

Rule to remember

After simplification, the equation must have the form ax^2 + bx + c = 0, with a ≠ 0 and no term of degree higher than 2.

Memory hook

Degree 2 only, not degree 1 or 3.

Examples and method

Worked example

Take 2x(x - 3) = 8. Expanding gives 2x^2 - 6x = 8. Bringing 8 to the left gives 2x^2 - 6x - 8 = 0. Since the highest power is 2, the equation is quadratic.

Method to apply

1. Expand brackets or remove fractions if needed. 2. Bring all terms to one side. 3. Check the highest power of x. 4. If the highest power is 2, it is quadratic.

Diagram support

A quick flow chart is useful: simplify first, check highest power, then classify the equation.

How CBSE asks it

Students may be asked whether a given relation is quadratic after expansion or simplification.

Avoid common mistakes

Common confusion

Students sometimes call any equation with x a quadratic equation, even when the highest power is 1 or 3.

Common wrong answer

Saying 3x + 5 = 0 is quadratic is wrong because the highest power is only 1.

Exam tip

Always simplify first. The equation is quadratic only if the highest power after simplification is exactly 2.

Quick check

Why is 2x(x - 3) = 8 a quadratic equation after simplification?

Because expanding gives 2x^2 - 6x - 8 = 0, and the highest power of x is 2. That is exactly the condition for a quadratic equation.

Answer writing and exam use

1-mark answer

A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0.

2-mark answer

A quadratic equation is any equation that can be simplified to ax^2 + bx + c = 0 with a ≠ 0. After simplification, the equation must have the form ax^2 + bx + c = 0, with a ≠ 0 and no term of degree higher than 2. 2x(x - 3) = 8 becomes 2x^2 - 6x - 8 = 0, so it is quadratic.

3-mark answer

Not every equation with x is quadratic. To identify a quadratic equation, first simplify it and check that the highest power of x is 2 and no higher power remains. After simplification, the equation must have the form ax^2 + bx + c = 0, with a ≠ 0 and no term of degree higher than 2. Take 2x(x - 3) = 8. Expanding gives 2x^2 - 6x = 8. Bringing 8 to the left gives 2x^2 - 6x - 8 = 0. Since the highest power is 2, the equation is quadratic. Students may be asked whether a given relation is quadratic after expansion or simplification. Saying 3x + 5 = 0 is quadratic is wrong because the highest power is only 1.
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