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Algebra and geometry · Level 10

Polynomials: zeroes

Zeroes of quadratics, their sum and product, and reading them off a graph.

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The idea

The zeros are where the curve touches or crosses the x-axis.

xy−3−2−112345−5−4−3−2−1123456x² − 2x − 3−13
xy−3−2−112345−5−4−3−2−1123456x² − 2x − 3−13

Look: This curve crosses the x-axis at −1 and 3.

✏️ Your turn!

Type your answer and press Check. Kutty will tell you if it is right.

  1. 1Zeros of x² − 9?

  2. 2Zeros of (x − 4)(x + 2)?

  3. 3Quadratic with zeros 1 and 4?

! Careful!

  • ✗Thinking every quadratic crosses the x-axis.

    ✓Some never do, like x² + 1.

  • ✗(x − 3) = 0 gives x = −3.

    ✓x − 3 = 0 gives x = 3.

See the whole lesson on one page

The zeros are where the curve touches or crosses the x-axis.

For a quadratic, there are at most two zeros. You can find them by factorising: set each bracket to 0.

Think of it like thisWater from a fountain goes up and comes down. Where it hits the ground (height 0) are the zeros.

xy−3−2−112345−5−4−3−2−1123456x² − 2x − 3−13
xy−3−2−112345−5−4−3−2−1123456x² − 2x − 3−13
This curve crosses the x-axis at −1 and 3.

Solved together

Question

Find the zeros of x² − 2x − 3.

  1. Find two numbers that multiply to −3 and add to −2: −3 and 1.

    (x − 3)(x + 1)

  2. Set each bracket to 0.

    x − 3 = 0 → x = 3x + 1 = 0 → x = −1

AnswerThe zeros are 3 and −1.

For older learners and grown-ups: a little more

For ax² + bx + c with zeros α and β: α + β = −b/a and αβ = c/a.

Now practise it

Kanakku makes new questions on polynomials: zeroes every time. It shows the working and gives a hint when you are stuck.

Play and practise