medium importancemedium difficulty8 min

Quadratic formula

The quadratic formula gives the roots of any quadratic equation ax² + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a.

In one line: Minus b, plus-minus D, over 2a.

Understand it

This formula is useful when factorisation is difficult or not obvious. Here a, b, and c are the coefficients from standard form, the expression under the square root is the discriminant, and the ± sign gives two possible roots.

Example

For 2x² - 3x - 2 = 0, the formula gives x = 2 or x = -1/2.

Formula and rule

x = (-b ± √(b² - 4ac)) / 2a.

Here a, b, c come from ax² + bx + c = 0, and the square-root part uses the discriminant.

See it worked out

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

Worked example

Solve 2x² - 3x - 2 = 0.

1Here a = 2, b = -3, c = -2.
2So x = (3 ± √((-3)² - 4(2)(-2))) / 4 = (3 ± √25) / 4.
3Therefore x = (3 ± 5) / 4, giving x = 2 and x = -1/2.

Method you can reuse

1Write the equation in standard form.
2Identify a, b, and c with signs.
3Compute D = b² - 4ac.
4Substitute into x = (-b ± √D) / 2a.
5Simplify carefully to get both roots.
6Check the answer if needed.

Visual tip

A substitution table with columns for a, b, c, then D, then x1 and x2, keeps the formula steps organised.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Why does the quadratic formula often give two answers for one equation?

Think first, then show answer

Because of the ± sign in the formula. That sign creates two possible values, so a quadratic equation usually gives two roots unless the discriminant is zero or negative.

Common wrong answer

Using b = 3 instead of b = -3 in 2x² - 3x - 2 = 0 gives the wrong roots because the sign is lost.

Don't mix this up

Students often write b with the wrong sign or forget that the denominator is 2a, not just 2.

Exam tip

First write the equation in standard form, then copy a, b, c carefully before substituting into the formula.

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 for roots, the nature of roots, or a full solution using the formula with correct substitution.

1
Enough for 1 mark
The quadratic formula gives the roots of any quadratic equation ax² + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a.
2
Add for 2 marks
Example: For 2x² - 3x - 2 = 0, the formula gives x = 2 or x = -1/2.
3
Add for 3 marks
This formula is useful when factorisation is difficult or not obvious.

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