Quadratic Factored Form Examples in Math

Start with the recap, study the fully worked examples, then use the practice problems to check your understanding of Quadratic Factored Form.

This page combines explanation, solved examples, and follow-up practice so you can move from recognition to confident problem-solving in Math.

Concept Recap

The factored form of a quadratic function is f(x) = a(x - r_1)(x - r_2), where r_1 and r_2 are the zeros (roots) of the function and a is the leading coefficient.

Each factor (x - r) equals zero when x = r. So the factored form literally shows you where the parabola crosses the x-axis—plug in either root and the whole expression becomes zero.

Read the full concept explanation →

How to Use These Examples

  • Read the first worked example with the solution open so the structure is clear.
  • Try the practice problems before revealing each solution.
  • Use the related concepts and background knowledge badges if you feel stuck.

What to Focus On

Core idea: Factored form directly reveals the zeros of the quadratic—the values where the function equals zero.

Common stuck point: Not every quadratic factors nicely over the integers—some have irrational or complex roots.

Sense of Study hint: Set each factor equal to zero and solve. The solutions are where the parabola crosses the x-axis.

Worked Examples

Example 1

easy
What are the zeros of f(x) = (x - 1)(x + 4)?

Solution

  1. 1
    Set each factor to zero: x - 1 = 0 gives x = 1; x + 4 = 0 gives x = -4.
  2. 2
    The zeros are x = 1 and x = -4.
  3. 3
    The graph crosses the x-axis at these points.

Answer

x = 1 \text{ and } x = -4
In factored form a(x - r_1)(x - r_2), the zeros are read directly as r_1 and r_2. This is the most convenient form for finding x-intercepts.

Example 2

medium
Write a quadratic in factored form with zeros at x = 3 and x = -2 and passing through (0, -12).

Practice Problems

Try these problems on your own first, then open the solution to compare your method.

Example 1

easy
Find the zeros of h(x) = -3(x + 2)(x - 5).

Example 2

medium
Find the axis of symmetry of f(x) = (x - 1)(x - 7).
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Background Knowledge

These ideas may be useful before you work through the harder examples.

quadratic functionsfactoring