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Standard form and coefficients

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

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Student-friendly explanation

To work correctly with quadratic equations, students must first write them in standard form. After that, the values of a, b, and c can be read directly, including their signs.

How to write this in exams

  1. 1

    Start with the exact idea

    The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

  2. 2

    Then show how to use it

    1. Write the equation as ax^2 + bx + c = 0. 2. Check the x^2 term first to get a. 3. Check the x term to get b, with sign. 4. Check the constant term to get c. 5. Use these values only after the equation is in standard form.

  3. 3

    Add one concrete example

    In 5x^2 - 2x + 9 = 0, a = 5, b = -2, and c = 9.

  4. 4

    Avoid this incomplete answer

    Writing b = 2 for 3x^2 - 2x + 9 = 0 is wrong because the coefficient of x is -2, not 2.

Definition

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

Example

In 5x^2 - 2x + 9 = 0, a = 5, b = -2, and c = 9.

Rule to remember

Standard form is ax^2 + bx + c = 0 with a ≠ 0. If a term is missing, its coefficient is 0, but the x^2 term must still be present.

Memory hook

a sits with x^2, b sits with x, c stands alone.

Examples and method

Worked example

For 3x^2 - 7x + 2 = 0, compare term by term with ax^2 + bx + c = 0. So a = 3, b = -7, and c = 2. These values are then used in factorisation, completing the square, or the quadratic formula.

Method to apply

1. Write the equation as ax^2 + bx + c = 0. 2. Check the x^2 term first to get a. 3. Check the x term to get b, with sign. 4. Check the constant term to get c. 5. Use these values only after the equation is in standard form.

Diagram support

A small coefficient table works well: x^2 term, x term, constant term, with a, b, c written below them.

How CBSE asks it

Students may be asked to identify coefficients, rewrite an equation in standard form, or use the right values in the quadratic formula.

Avoid common mistakes

Common confusion

Students often forget that the sign of b or c belongs to the coefficient, not just the number written after it.

Common wrong answer

Writing b = 2 for 3x^2 - 2x + 9 = 0 is wrong because the coefficient of x is -2, not 2.

Exam tip

Always compare the given equation term by term with ax^2 + bx + c = 0 before using any formula.

Quick check

What are a, b, and c in x^2 + 7x - 8 = 0?

Here a = 1, b = 7, and c = -8. The coefficient of x^2 is 1 even when it is not written, and the sign of the constant term must be kept exactly as given.

Answer writing and exam use

1-mark answer

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0.

2-mark answer

The standard form of a quadratic equation is ax^2 + bx + c = 0, where a, b, and c are coefficients and a ≠ 0. Standard form is ax^2 + bx + c = 0 with a ≠ 0. If a term is missing, its coefficient is 0, but the x^2 term must still be present. In 5x^2 - 2x + 9 = 0, a = 5, b = -2, and c = 9.

3-mark answer

To work correctly with quadratic equations, students must first write them in standard form. After that, the values of a, b, and c can be read directly, including their signs. Standard form is ax^2 + bx + c = 0 with a ≠ 0. If a term is missing, its coefficient is 0, but the x^2 term must still be present. For 3x^2 - 7x + 2 = 0, compare term by term with ax^2 + bx + c = 0. So a = 3, b = -7, and c = 2. These values are then used in factorisation, completing the square, or the quadratic formula. Students may be asked to identify coefficients, rewrite an equation in standard form, or use the right values in the quadratic formula. Writing b = 2 for 3x^2 - 2x + 9 = 0 is wrong because the coefficient of x is -2, not 2.
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