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The zero product rule

The zero product rule is based on the fact that a product is zero only if at least one factor is zero. (x − 2)(x + 5) = 0 gives x − 2 = 0 or x + 5 = 0, so x = 2 or x = −5.

Maths 2 · Updated

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Why does it work?

If one factor is zero, the product is zero: 7 · 0 = 0. If no factor is zero, the product is never zero. So if the product is zero, at least one factor is zero.

(x − 4)(x + 3) = 0 gives x − 4 = 0 or x + 3 = 0

Zero on one side

We want zero on one side and a product on the other.

(x − 1)(x + 2) = 4 cannot be solved directly, because 4 can be many different products. First rewrite it as x² + x − 6 = 0 and factorise: (x + 3)(x − 2) = 0.

  • Make one side zero by subtracting from both sides.
  • Write the other side as a product, for example by taking out x.
  • Set each factor equal to zero and solve.
  • Check each solution in the original equation.

Take out x

If there is no constant term, x is in every term. Then we take out x.

x² − 7x = 0 gives x(x − 7) = 0

That gives x = 0 or x = 7. The solution x = 0 is often forgotten.

Example

Example 1: (x − 4)(x + 3) = 0

We want to find the values of x that make the product zero. One side is already zero.

  1. At least one factor must be zero.

    x − 4 = 0 or x + 3 = 0

  2. In the first one, we add 4 to both sides.

    x = 4

  3. In the second one, we subtract 3 from both sides.

    x = −3

  4. The equation has two solutions.

    x = 4 or x = −3

x = 4: (4 − 4)(4 + 3) = 0 · 7 = 0. x = −3: (−7) · 0 = 0. Both give zero, so both are correct.

Example 2: 2x² − 10x = 0

We want to write the left-hand side as a product. Both terms contain 2x, so we take out 2x.

  1. 2x² = 2x · x and 10x = 2x · 5.

    2x(x − 5) = 0

  2. Check the factorisation: 2x · x = 2x² and 2x · (−5) = −10x. That is correct.

  3. We set each factor equal to zero.

    2x = 0 or x − 5 = 0

  4. We divide the first one by 2 and add 5 in the second.

    x = 0 or x = 5

x = 0: 2 · 0 − 10 · 0 = 0. x = 5: 2 · 25 − 10 · 5 = 50 − 50 = 0. Both are correct.

Example 3: x² = 6x

We want zero on one side. We do not divide by x, because then the solution x = 0 disappears.

  1. We subtract 6x from both sides.

    x² − 6x = 0

  2. Both terms contain x. We take out x.

    x(x − 6) = 0

  3. One of the factors must be zero.

    x = 0 or x − 6 = 0

  4. In the second one, we add 6 to both sides.

    x = 0 or x = 6

x = 0: 0² = 0 and 6 · 0 = 0. x = 6: 6² = 36 and 6 · 6 = 36. Both are solutions.

Pitfalls

  • Dividing both sides by x, so that x² = 6x only gives x = 6.

    x can be zero, and we cannot divide by zero. Collect everything on one side and take out x instead.

  • Using the rule when the right-hand side is not zero, as in (x − 1)(x + 2) = 4.

    Multiply out and subtract 4: x² + x − 6 = 0. Factorise: (x + 3)(x − 2) = 0, so x = −3 or x = 2.

  • Getting the sign wrong, so that the factor (x + 3) gives x = 3.

    Solve x + 3 = 0: subtract 3 from both sides, x = −3. Try it: −3 + 3 = 0.

Practise on your own

0 of 4 correct

  1. Solve the equation (x − 6)(x + 2) = 0.

  2. Solve the equation x² + 9x = 0.

  3. Solve the equation (2x − 3)(x + 1) = 0.

  4. Solve the equation x(x − 3) = 10.

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Common questions

When should I use the zero product rule and when the pq formula?

The zero product rule fits when the equation is a product equal to zero, or when you can take out x. Otherwise the pq formula is usually the most convenient.

Does the rule work with more than two factors?

Yes. x(x − 1)(x + 4) = 0 gives x = 0, x = 1 or x = −4. Each factor gives one solution.

What does the zero product rule have to do with the graph?

The solutions are the values of x where the graph crosses the x-axis, that is, where y is zero. They are called zeros.

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