Zeros of a Quadratic Examples in Math

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

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 zeros (or roots) of a quadratic function f(x) = ax^2 + bx + c are the values of x where f(x) = 0. Graphically, they are the x-intercepts of the parabola.

The zeros are where the parabola crosses or touches the x-axis. A parabola can cross twice (two zeros), just touch once (one repeated zero), or miss entirely (no real zeros). You can find them by factoring, completing the square, or using the quadratic formula.

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: Zeros are the bridge between algebra and geometry—they are both the solutions to the equation f(x) = 0 and the points where the graph meets the x-axis.

Common stuck point: Choosing the right method: try factoring first; if that fails, use the quadratic formula. The discriminant tells you in advance how many real zeros to expect.

Sense of Study hint: Try factoring first. If you cannot find integer factors within 30 seconds, switch to the quadratic formula.

Worked Examples

Example 1

easy
Find the zeros of f(x) = x^2 - 7x + 10.

Solution

  1. 1
    Set f(x) = 0: x^2 - 7x + 10 = 0.
  2. 2
    Factor: (x - 2)(x - 5) = 0.
  3. 3
    Zeros: x = 2 and x = 5.

Answer

x = 2 \text{ and } x = 5
The zeros of a function are the x-values where f(x) = 0. They correspond to the x-intercepts of the graph.

Example 2

medium
Find the zeros of g(x) = 2x^2 - 8x.

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) = x^2 - 1.

Example 2

hard
How many zeros does f(x) = x^2 + 4 have over the reals?
← Back to Zeros of a Quadratic Formula Explained Practice Mode

Background Knowledge

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

quadratic functionsfactoringquadratic formula