medium difficulty5 min

Quadratic polynomial and its zeros

A quadratic polynomial is a polynomial of degree 2, usually written as ax²+bx+c where a is not zero. Its zeros are the values of x that make the polynomial equal to 0.

In one line: Quadratic means up to two zeros.

Practice This Concept

Understand it

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 ax²+bx+c=0 and to connect those values with the graph.

Example

For x²-5x+6, the zeros are 2 and 3 because (x-2)(x-3)=0.

Formula and rule

For ax²+bx+c, zeros are the x-values satisfying ax²+bx+c=0.

A quadratic polynomial has at most two zeros.

See it worked out

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

Worked example

1For x²-7x+12, factorise as (x-3)(x-4)=0.
2So the zeros are 3 and 4.

Method you can reuse

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

Visual tip

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

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

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

Think first, then show answer

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.

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.

Don't mix this up

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

Exam tip

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