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Quadratic polynomial and its zeros

A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

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Student-friendly explanation

A quadratic polynomial can have either two real zeros, one repeated zero, or no real zero, depending on the graph and the discriminant in later chapters. In Class 10 polynomials, the main idea is to find the values of x that satisfy ax^2+bx+c=0 and to connect those values with the graph.

How to write this in exams

  1. 1

    Start with the exact idea

    A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

  2. 2

    Then show how to use it

    1. Write the polynomial equal to zero. 2. Factorise if possible. 3. Set each factor equal to zero. 4. Solve for x and verify by substitution.

  3. 3

    Add one concrete example

    For x^2-5x+6, the zeros are 2 and 3 because (x-2)(x-3)=0.

  4. 4

    Avoid this incomplete answer

    A common wrong answer is to take only one factor as the answer. Both factors must be used because each can give a zero.

Definition

A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

Example

For x^2-5x+6, the zeros are 2 and 3 because (x-2)(x-3)=0.

Rule to remember

For ax^2+bx+c, zeros are the x-values satisfying ax^2+bx+c=0. A quadratic polynomial has at most two zeros.

Memory hook

Quadratic means up to two zeros.

Examples and method

Worked example

For x^2-7x+12, factorise as (x-3)(x-4)=0. So the zeros are 3 and 4.

Method to apply

1. Write the polynomial equal to zero. 2. Factorise if possible. 3. Set each factor equal to zero. 4. Solve for x and verify by substitution.

Diagram support

The graph is a parabola. The zeros are the x-coordinates where the parabola cuts or touches the x-axis.

How CBSE asks it

Find the zeros of a quadratic polynomial, verify them, or relate them to the graph.

Avoid common mistakes

Common confusion

Students sometimes think a quadratic must always have two different zeros. Repeated roots are also possible.

Common wrong answer

A common wrong answer is to take only one factor as the answer. Both factors must be used because each can give a zero.

Exam tip

If a quadratic factorises neatly, first break it into factors and then set each factor equal to zero.

Quick check

What are the zeros of a quadratic polynomial used for in this chapter?

The zeros are the values of x that make the quadratic polynomial equal to zero. They help us solve the equation, factorise the polynomial, and understand its graph.

Study the quadratic polynomial and its zeros diagram carefully

Use the labelled diagram to keep quadratic polynomial and its zeros clear in short answers and revision.

What this diagram makes clear

This diagram keeps the labels and direction of quadratic polynomial and its zeros in the right order.

Where this helps in exams

Use this for labelled diagram work and short exam answers on quadratic polynomial and its zeros.

Revision cue

Revise quadratic polynomial and its zeros through the labels before writing the answer.

Answer writing and exam use

1-mark answer

A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

2-mark answer

A quadratic polynomial is a polynomial of degree 2, usually written as ax^2+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0. For ax^2+bx+c, zeros are the x-values satisfying ax^2+bx+c=0. A quadratic polynomial has at most two zeros. For x^2-5x+6, the zeros are 2 and 3 because (x-2)(x-3)=0.

3-mark answer

A quadratic polynomial can have either two real zeros, one repeated zero, or no real zero, depending on the graph and the discriminant in later chapters. In Class 10 polynomials, the main idea is to find the values of x that satisfy ax^2+bx+c=0 and to connect those values with the graph. For ax^2+bx+c, zeros are the x-values satisfying ax^2+bx+c=0. A quadratic polynomial has at most two zeros. For x^2-7x+12, factorise as (x-3)(x-4)=0. So the zeros are 3 and 4. Find the zeros of a quadratic polynomial, verify them, or relate them to the graph. A common wrong answer is to take only one factor as the answer. Both factors must be used because each can give a zero.
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