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Quadratic formula

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

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Student-friendly explanation

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.

How to write this in exams

  1. 1

    Start with the exact idea

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

  2. 2

    Then show how to use it

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

  3. 3

    Add one concrete example

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

  4. 4

    Avoid this incomplete answer

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

Definition

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

Example

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

Rule to remember

x = (-b ± √(b² - 4ac)) / 2a. Here a, b, c come from ax^2 + bx + c = 0, and the square-root part uses the discriminant.

Memory hook

Minus b, plus-minus D, over 2a.

Examples and method

Worked example

Solve 2x^2 - 3x - 2 = 0. Here a = 2, b = -3, c = -2. So x = (3 ± √((-3)^2 - 4(2)(-2))) / 4 = (3 ± √25) / 4. Therefore x = (3 ± 5) / 4, giving x = 2 and x = -1/2.

Method to apply

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

Diagram support

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

How CBSE asks it

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

Avoid common mistakes

Common confusion

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

Common wrong answer

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

Exam tip

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

Quick check

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

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.

Answer writing and exam use

1-mark answer

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

2-mark answer

The quadratic formula gives the roots of any quadratic equation ax^2 + bx + c = 0 as x = (-b ± √(b² - 4ac)) / 2a. x = (-b ± √(b² - 4ac)) / 2a. Here a, b, c come from ax^2 + bx + c = 0, and the square-root part uses the discriminant. For 2x^2 - 3x - 2 = 0, the formula gives x = 2 or x = -1/2.

3-mark answer

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. x = (-b ± √(b² - 4ac)) / 2a. Here a, b, c come from ax^2 + bx + c = 0, and the square-root part uses the discriminant. Solve 2x^2 - 3x - 2 = 0. Here a = 2, b = -3, c = -2. So x = (3 ± √((-3)^2 - 4(2)(-2))) / 4 = (3 ± √25) / 4. Therefore x = (3 ± 5) / 4, giving x = 2 and x = -1/2. CBSE may ask for roots, the nature of roots, or a full solution using the formula with correct substitution. Using b = 3 instead of b = -3 in 2x^2 - 3x - 2 = 0 gives the wrong roots because the sign is lost.
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